Overview of More observations Complex Networks Basic definitions Examples of Complex Networks ◮ But surely networks aren’t new... Properties of Complex Networks ◮ Graph theory is well established... Nutshell ◮ Study of social networks started in the 1930’s... Basic models of complex networks ◮ So why all this ‘new’ research on networks? Generalized random networks ◮ Answer: Oodles of Easily Accessible Data. Scale-free networks Small-world networks Generalized affiliation ◮ We can now inform (alas) our theories networks with a much more measurable reality. ∗ References ◮ A worthy goal: establish mechanistic explanations. ∗ If this is upsetting, maybe string theory is for you... 15 of 128
Overview of More observations Complex Networks Basic definitions Examples of ◮ Web-scale data sets can be overly exciting. Complex Networks Properties of Complex Networks Witness: Nutshell ◮ The End of Theory: The Data Deluge Makes the Basic models of complex networks Scientific Theory Obsolete (Anderson, Wired) ( ⊞ ) Generalized random networks ◮ “The Unreasonable Effectiveness of Data,” Scale-free networks Small-world networks Halevy et al. [15] . Generalized affiliation networks References But: ◮ For scientists, description is only part of the battle. ◮ We still need to understand. 16 of 128
Overview of Super Basic definitions Complex Networks Basic definitions Examples of Complex Networks Nodes = A collection of entities which have Properties of properties that are somehow related to each other Complex Networks Nutshell ◮ e.g., people, forks in rivers, proteins, webpages, Basic models of organisms,... complex networks Generalized random networks Scale-free networks Small-world networks Links = Connections between nodes Generalized affiliation networks ◮ Links may be directed or undirected. References ◮ Links may be binary or weighted. Other spiffing words: vertices and edges. 17 of 128
Overview of Super Basic definitions Complex Networks Basic definitions Examples of Node degree = Number of links per node Complex Networks Properties of Complex Networks ◮ Notation: Node i ’s degree = k i . Nutshell ◮ k i = 0,1,2,. . . . Basic models of complex networks ◮ Notation: the average degree of a network = � k � Generalized random networks (and sometimes z ) Scale-free networks Small-world networks ◮ Connection between number of edges m and Generalized affiliation networks average degree: References � k � = 2 m N . ◮ Defn: N i = the set of i ’s k i neighbors 18 of 128
Overview of Super Basic definitions Complex Networks Basic definitions Examples of Adjacency matrix: Complex Networks Properties of ◮ We represent a directed network by a matrix A with Complex Networks link weight a ij for nodes i and j in entry ( i , j ) . Nutshell Basic models of ◮ e.g., complex networks Generalized random 0 1 1 1 0 networks Scale-free networks 0 0 1 0 1 Small-world networks Generalized affiliation A = 1 0 0 0 0 networks 0 1 0 0 1 References 0 1 0 1 0 ◮ (n.b., for numerical work, we always use sparse matrices.) 19 of 128
Overview of Examples Complex Networks Basic definitions Examples of Comple Properties of Complex Networks So what passes for a complex network? Nutshell Basic models of ◮ Complex networks are large (in node number) complex networks Generalized random ◮ Complex networks are sparse (low edge to node networks Scale-free networks ratio) Small-world networks Generalized affiliation networks ◮ Complex networks are usually dynamic and evolving References ◮ Complex networks can be social, economic, natural, informational, abstract, ... 20 of 128
Overview of Examples Complex Networks Basic definitions Physical networks Examples of Comple ◮ River networks Properties of ◮ The Internet Complex Networks ◮ Neural networks ◮ Road networks Nutshell ◮ Trees and leaves Basic models of ◮ Power grids complex networks ◮ Blood networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References ◮ Distribution (branching) versus redistribution (cyclical) 21 of 128
Overview of Examples Complex Networks Interaction networks Basic definitions Examples of Comple ◮ The Blogosphere Properties of Complex Networks ◮ Biochemical Nutshell networks Basic models of ◮ Gene-protein complex networks Generalized random networks networks Scale-free networks Small-world networks ◮ Food webs: who Generalized affiliation networks eats whom References ◮ The World Wide Web (?) ◮ Airline networks ◮ Call networks datamining.typepad.com ( ⊞ ) (AT&T) ◮ The Media 22 of 128
Overview of Examples Complex Networks Basic definitions Interaction networks: Examples of Comple social networks Properties of Complex Networks ◮ Snogging Nutshell ◮ Friendships Basic models of complex networks ◮ Acquaintances Generalized random networks Scale-free networks ◮ Boards and Small-world networks Generalized affiliation directors networks References ◮ Organizations ◮ facebook ( ⊞ ) twitter ( ⊞ ), (Bearman et al. , 2004) ◮ ‘Remotely sensed’ by: email activity, instant messaging, phone logs (*cough*). 23 of 128
Overview of Examples Complex Networks Basic definitions Examples of Comple Properties of Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References 24 of 128
Overview of Examples Complex Networks Relational networks Basic definitions ◮ Consumer purchases Examples of Comple (Wal-Mart: ≈ 1 petabyte = 10 15 bytes) Properties of Complex Networks ◮ Thesauri: Networks of words generated by meanings Nutshell ◮ Knowledge/Databases/Ideas Basic models of complex networks ◮ Metadata—Tagging: del.icio.us ( ⊞ ) flickr ( ⊞ ) Generalized random networks Scale-free networks Small-world networks Generalized affiliation common tags cloud | list networks References community daily dictionary education encyclopedia english free imported info information internet knowledge news reference research learning resource web2.0 wiki resources search tools useful web wikipedia 25 of 128
Overview of Clickworthy Science: Complex Networks Basic definitions Examples of Comple Properties of Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References Bollen et al. [7] 26 of 128
Overview of A notable feature of large-scale networks: Complex Networks ◮ Graphical renderings are often just a big mess. Basic definitions Examples of Complex Networks Properties of Comple ⇐ Typical hairball Nutshell Basic models of ◮ number of nodes N = 500 complex networks Generalized random networks ◮ number of edges m = 1000 Scale-free networks Small-world networks ◮ average degree � k � = 4 Generalized affiliation networks References ◮ And even when renderings somehow look good: “That is a very graphic analogy which aids understanding wonderfully while being, strictly speaking, wrong in every possible way” said Ponder [Stibbons] — Making Money , T. Pratchett. ◮ We need to extract digestible, meaningful aspects. 27 of 128
Overview of Properties Complex Networks Basic definitions Examples of Some key features of real complex networks: Complex Networks Properties of Comple ◮ Degree ◮ Concurrency Nutshell distribution ◮ Hierarchical Basic models of complex networks ◮ Assortativity scaling Generalized random networks ◮ Homophily ◮ Network distances Scale-free networks Small-world networks Generalized affiliation ◮ Clustering ◮ Centrality networks References ◮ Motifs ◮ Efficiency ◮ Modularity ◮ Robustness ◮ Coevolution of network structure and processes on networks. 28 of 128
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks 1. Degree distribution P k Properties of Comple Nutshell ◮ P k is the probability that a randomly selected node Basic models of has degree k complex networks Generalized random ◮ Big deal: Form of P k key to network’s behavior networks Scale-free networks Small-world networks ◮ ex 1: Erd˝ os-Rényi random networks have a Poisson Generalized affiliation networks distribution: References P k = e −� k � � k � k / k ! ◮ ex 2: “Scale-free” networks: P k ∝ k − γ ⇒ ‘hubs’ ◮ We’ll come back to this business soon... 29 of 128
Overview of Properties Complex Networks Basic definitions 2. Assortativity/3. Homophily: Examples of Complex Networks Properties of Comple ◮ Social networks: Homophily ( ⊞ ) = birds of a feather Nutshell ◮ e.g., degree is standard property for sorting: Basic models of complex networks measure degree-degree correlations. Generalized random ◮ Assortative network: [20] similar degree nodes networks Scale-free networks Small-world networks connecting to each other. Generalized affiliation networks ◮ Often social: company directors, coauthors, actors. References ◮ Disassortative network: high degree nodes connecting to low degree nodes. ◮ Often technological or biological: Internet, protein interactions, neural networks, food webs. 30 of 128
Overview of Properties Complex Networks Basic definitions 4. Clustering: Examples of Complex Networks ◮ Your friends tend to know each other. Properties of Comple ◮ Two measures: Nutshell Basic models of complex networks �� � j 1 j 2 ∈N i a j 1 j 2 Generalized random due to Watts & Strogatz [30] C 1 = networks k i ( k i − 1 ) / 2 Scale-free networks Small-world networks i Generalized affiliation networks C 2 = 3 × # triangles References due to Newman [21] # triples ◮ C 1 is the average fraction of pairs of neighbors who are connected. ◮ Interpret C 2 as probability two of a node’s friends know each other. 31 of 128
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks Properties of Comple 5. Motifs: Nutshell ◮ Small, recurring functional subnetworks Basic models of complex networks ◮ e.g., Feed Forward Loop: Generalized random networks Scale-free networks a feedforward loop Small-world networks X Generalized affiliation X networks Y Y References n Z Shen-Orr, Uri Alon, et al. [23] 32 of 128
Overview of Properties Complex Networks 6. modularity: Basic definitions Examples of Complex Networks 49 53 58 Properties of Comple 63 46 83 114 28 33 11 Nutshell 25 97 88 1 59 67 73 105 24 Basic models of 50 103 37 complex networks 89 69 36 45 110 109 57 90 Generalized random 44 66 34 networks 42 16 75 82 4 Scale-free networks 31 93 86 91 112 80 Small-world networks 0 48 18 54 9 92 Generalized affiliation networks 23 7 29 104 8 61 71 94 41 35 78 68 References 99 22 19 55 21 77 5 10 111 30 81 101 79 3 108 51 85 38 52 84 98 113 2 6 17 76 43 26 70 107 60 39 40 14 74 72 47 62 95 96 12 13 27 100 15 102 65 20 87 106 56 64 32 Clauset et al. , 2006 [10] : NCAA football 33 of 128
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks Properties of Comple 7. Concurrency: Nutshell Basic models of ◮ Transmission of a contagious element only occurs complex networks during contact [18] Generalized random networks Scale-free networks ◮ Rather obvious but easily missed in a simple model Small-world networks Generalized affiliation networks ◮ Dynamic property—static networks are not enough References ◮ Knowledge of previous contacts crucial ◮ Beware cumulated network data! 34 of 128
Overview of Properties Complex Networks 8. Horton-Strahler stream ordering: Basic definitions Examples of ◮ Metrics for branching networks: Complex Networks ◮ Method for ordering streams hierarchically Properties of Comple ◮ Reveals fractal nature of natural branching networks Nutshell ◮ Hierarchy is not pure but mixed (Tokunaga). [25, 12] Basic models of ◮ Major examples: rivers and blood networks. complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References (c) (b) (a) ◮ Beautifully described but poorly explained. 35 of 128
Overview of Properties Complex Networks Basic definitions Examples of 9. Network distances: Complex Networks Properties of Comple (a) shortest path length d ij : Nutshell Basic models of ◮ Fewest number of steps between nodes i and j . complex networks Generalized random ◮ (Also called the chemical distance between i and j .) networks Scale-free networks Small-world networks Generalized affiliation networks (b) average path length � d ij � : References ◮ Average shortest path length in whole network. ◮ Good algorithms exist for calculation. ◮ Weighted links can be accommodated. 36 of 128
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks 9. Network distances: Properties of Comple (c) Network diameter d max : Nutshell Basic models of ◮ Maximum shortest path length in network. complex networks Generalized random networks Scale-free networks Small-world networks ij d − 1 � n � ] − 1 : (d) Closeness d cl = [ � / Generalized affiliation ij 2 networks References ◮ Average ‘distance’ between any two nodes. ◮ Closeness handles disconnected networks ( d ij = ∞ ) ◮ d cl = ∞ only when all nodes are isolated. 37 of 128
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks 10. Centrality: Properties of Comple Nutshell ◮ Many such measures of a node’s ‘importance.’ Basic models of ◮ ex 1: Degree centrality: k i . complex networks Generalized random networks ◮ ex 2: Node i ’s betweenness Scale-free networks Small-world networks = fraction of shortest paths that pass through i . Generalized affiliation networks ◮ ex 3: Edge ℓ ’s betweenness References = fraction of shortest paths that travel along ℓ . ◮ ex 4: Recursive centrality: Hubs and Authorities (Jon Kleinberg [17] ) 38 of 128
Overview of Nutshell: Complex Networks Basic definitions Overview Key Points: Examples of Complex Networks ◮ The field of complex networks came into existence in Properties of Complex Networks the late 1990s. Nutshell ◮ Explosion of papers and interest since 1998/99. Basic models of complex networks ◮ Hardened up much thinking about complex systems. Generalized random networks Scale-free networks ◮ Specific focus on networks that are large-scale, Small-world networks Generalized affiliation sparse, natural or man-made, evolving and dynamic, networks References and (crucially) measurable. ◮ Three main (blurred) categories: 1. Physical (e.g., river networks), 2. Interactional (e.g., social networks), 3. Abstract (e.g., thesauri). 39 of 128
Overview of Nutshell: Complex Networks Basic definitions Overview Key Points (cont.): Examples of Complex Networks ◮ Obvious connections with the vast extant field of Properties of Complex Networks graph theory. Nutshell ◮ But focus on dynamics is more of a Basic models of physics/stat-mech/comp-sci flavor. complex networks Generalized random ◮ Two main areas of focus: networks Scale-free networks Small-world networks 1. Description: Characterizing very large networks Generalized affiliation networks 2. Explanation: Micro story ⇒ Macro features References ◮ Some essential structural aspects are understood: degree distribution, clustering, assortativity, group structure, overall structure,... ◮ Still much work to be done, especially with respect to dynamics... 40 of 128
Overview of Models Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks Some important models: Nutshell 1. generalized random networks Basic models of comple Generalized random networks 2. scale-free networks Scale-free networks Small-world networks 3. small-world networks Generalized affiliation networks 4. statistical generative models ( p ∗ ) References 5. generalized affiliation networks 41 of 128
Overview of Models Complex Networks Basic definitions Examples of Complex Networks Properties of Generalized random networks: Complex Networks Nutshell ◮ Arbitrary degree distribution P k . Basic models of complex networks ◮ Create (unconnected) nodes with degrees sampled Generalized random networks Scale-free networks from P k . Small-world networks Generalized affiliation ◮ Wire nodes together randomly. networks References ◮ Create ensemble to test deviations from randomness. 43 of 128
Overview of Building random networks: Stubs Complex Networks Phase 1: Basic definitions Examples of ◮ Idea: start with a soup of unconnected nodes with Complex Networks stubs (half-edges): Properties of Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks ◮ Randomly select stubs References (not nodes!) and connect them. ◮ Must have an even number of stubs. ◮ Initially allow self- and repeat connections. 44 of 128
Overview of Building random networks: First rewiring Complex Networks Basic definitions Examples of Complex Networks Phase 2: Properties of Complex Networks ◮ Now find any (A) self-loops and (B) repeat edges and Nutshell randomly rewire them. Basic models of complex networks Generalized random networks Scale-free networks Small-world networks (A) (B) Generalized affiliation networks ◮ Being careful: we can’t change the degree of any References node, so we can’t simply move links around. ◮ Simplest solution: randomly rewire two edges at a time. 45 of 128
Overview of General random rewiring algorithm Complex Networks i 2 e 1 i 1 Basic definitions ◮ Randomly choose two edges. Examples of Complex Networks (Or choose problem edge and Properties of Complex Networks a random edge) Nutshell ◮ Check to make sure edges Basic models of are disjoint. complex networks Generalized random networks i 4 e i 3 Scale-free networks 2 Small-world networks Generalized affiliation i 2 networks i References 1 ◮ Rewire one end of each edge. ◮ Node degrees do not change. e’ ◮ Works if e 1 is a self-loop or e’ 2 1 repeated edge. ◮ Same as finding on/off/on/off 4-cycles. and rotating them. i 4 i 3 46 of 128
Overview of Sampling random networks Complex Networks Basic definitions Examples of Complex Networks Phase 2: Properties of Complex Networks ◮ Use rewiring algorithm to remove all self and repeat Nutshell loops. Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Phase 3: Generalized affiliation networks ◮ Randomize network wiring by applying rewiring References algorithm liberally. ◮ Rule of thumb: # Rewirings ≃ 10 × # edges [19] . 47 of 128
Overview of Scale-free networks Complex Networks Basic definitions Examples of ◮ Networks with power-law degree distributions have Complex Networks become known as scale-free networks. Properties of Complex Networks ◮ Scale-free refers specifically to the degree Nutshell distribution having a power-law decay in its tail: Basic models of complex networks Generalized random P k ∼ k − γ for ‘large’ k networks Scale-free networks Small-world networks Generalized affiliation networks References ◮ One of the seminal works in complex networks: Laszlo Barabási and Reka Albert, Science, 1999: “Emergence of scaling in random networks” [4] ◮ Somewhat misleading nomenclature... 49 of 128
Overview of Scale-free networks Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks ◮ Scale-free networks are not fractal in any sense. Nutshell ◮ Usually talking about networks whose links are Basic models of complex networks abstract, relational, informational, . . . (non-physical) Generalized random networks Scale-free networks ◮ Primary example: hyperlink network of the Web Small-world networks Generalized affiliation ◮ Much arguing about whether or networks are networks References ‘scale-free’ or not. . . 50 of 128
Overview of Random networks: largest components Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks Nutshell Basic models of complex networks γ = 2.5 γ = 2.5 γ = 2.5 γ = 2.5 Generalized random � k � = 1.8 � k � = 2.05333 � k � = 1.66667 � k � = 1.92 networks Scale-free networks Small-world networks Generalized affiliation networks References γ = 2.5 γ = 2.5 γ = 2.5 γ = 2.5 � k � = 1.6 � k � = 1.50667 � k � = 1.62667 � k � = 1.8 51 of 128
Overview of Scale-free networks Complex Networks Basic definitions Examples of Complex Networks The big deal: Properties of Complex Networks ◮ We move beyond describing networks to finding Nutshell mechanisms for why certain networks are the way Basic models of complex networks they are. Generalized random networks Scale-free networks Small-world networks A big deal for scale-free networks: Generalized affiliation networks References ◮ How does the exponent γ depend on the mechanism? ◮ Do the mechanism details matter? 52 of 128
Overview of BA model Complex Networks Basic definitions Examples of ◮ Barabási-Albert model = BA model. Complex Networks Properties of ◮ Key ingredients: Complex Networks Growth and Preferential Attachment (PA). Nutshell ◮ Step 1: start with m 0 disconnected nodes. Basic models of complex networks ◮ Step 2: Generalized random networks Scale-free networks 1. Growth—a new node appears at each time step Small-world networks Generalized affiliation t = 0 , 1 , 2 , . . . . networks 2. Each new node makes m links to nodes already References present. 3. Preferential attachment—Probability of connecting to i th node is ∝ k i . ◮ In essence, we have a rich-gets-richer scheme. 53 of 128
Overview of BA model Complex Networks Basic definitions ◮ Definition: A k is the attachment kernel for a node Examples of Complex Networks with degree k . Properties of ◮ For the original model: Complex Networks Nutshell A k = k Basic models of complex networks Generalized random networks ◮ Definition: P attach ( k , t ) is the attachment probability. Scale-free networks Small-world networks ◮ For the original model: Generalized affiliation networks References k i ( t ) k i ( t ) P attach ( node i , t ) = = � N ( t ) � k max ( t ) j = 1 k j ( t ) kN k ( t ) k = 0 where N ( t ) = m 0 + t is # nodes at time t and N k ( t ) is # degree k nodes at time t . 54 of 128
Overview of Approximate analysis Complex Networks ◮ When ( N + 1 ) th node is added, the expected Basic definitions increase in the degree of node i is Examples of Complex Networks k i , N Properties of E ( k i , N + 1 − k i , N ) ≃ m . Complex Networks � N ( t ) j = 1 k j ( t ) Nutshell Basic models of complex networks ◮ Assumes probability of being connected to is small. Generalized random networks ◮ Dispense with Expectation by assuming (hoping) that Scale-free networks Small-world networks over longer time frames, degree growth will be Generalized affiliation networks smooth and stable. References ◮ Approximate k i , N + 1 − k i , N with d d t k i , t : k i ( t ) d d t k i , t = m � N ( t ) j = 1 k j ( t ) where t = N ( t ) − m 0 . 55 of 128
Overview of Approximate analysis Complex Networks ◮ Deal with denominator: each added node brings m Basic definitions new edges. Examples of N ( t ) Complex Networks � k j ( t ) = 2 tm ∴ Properties of Complex Networks j = 1 Nutshell Basic models of ◮ The node degree equation now simplifies: complex networks Generalized random networks 2 mt = 1 d k i ( t ) = mk i ( t ) Scale-free networks d t k i , t = m 2 t k i ( t ) Small-world networks � N ( t ) Generalized affiliation j = 1 k j ( t ) networks References ◮ Rearrange and solve: d k i ( t ) k i ( t ) = d t 2 t ⇒ k i ( t ) = c i t 1 / 2 . ◮ Next find c i . . . 56 of 128
Overview of Approximate analysis Complex Networks Basic definitions Examples of ◮ Know i th node appears at time Complex Networks � i − m 0 Properties of for i > m 0 Complex Networks t i , start = for i ≤ m 0 Nutshell 0 Basic models of complex networks ◮ So for i > m 0 (exclude initial nodes), we must have Generalized random networks Scale-free networks Small-world networks � 1 / 2 � t Generalized affiliation for t ≥ t i , start . k i ( t ) = m networks t i , start References ◮ All node degrees grow as t 1 / 2 but later nodes have larger t i , start which flattens out growth curve. ◮ Early nodes do best (First-mover advantage). 57 of 128
Overview of Approximate analysis Complex Networks Basic definitions 20 Examples of Complex Networks Properties of Complex Networks 15 Nutshell Basic models of k i (t) complex networks ◮ m = 3 Generalized random 10 networks Scale-free networks ◮ t i , start = Small-world networks Generalized affiliation 1 , 2 , 5 , and 10. networks 5 References 0 0 10 20 30 40 50 t 58 of 128
Overview of Degree distribution Complex Networks Basic definitions ◮ So what’s the degree distribution at time t ? Examples of ◮ Use fact that birth time for added nodes is distributed Complex Networks Properties of uniformly: Complex Networks Pr ( t i , start ) d t i , start ≃ d t i , start Nutshell t Basic models of complex networks ◮ Also use Generalized random networks Scale-free networks � 1 / 2 ⇒ t i , start = m 2 t Small-world networks � t Generalized affiliation k i ( t ) = m k i ( t ) 2 . networks t i , start References Transform variables—Jacobian: = − 2 m 2 t d t i , start k i ( t ) 3 . d k i 59 of 128
Overview of Degree distribution Complex Networks Basic definitions ◮ Examples of Pr ( k i ) d k i = Pr ( t i , start ) d t i , start Complex Networks Properties of ◮ Complex Networks � � d t i , start � � Nutshell = Pr ( t i , start ) d k i � � d k i Basic models of � � complex networks ◮ Generalized random networks t d k i 2 m 2 t = 1 Scale-free networks Small-world networks k i ( t ) 3 Generalized affiliation networks ◮ References = 2 m 2 k i ( t ) 3 d k i ◮ ∝ k − 3 d k i . i 60 of 128
Overview of Degree distribution Complex Networks Basic definitions Examples of Complex Networks ◮ We thus have a very specific prediction of Properties of Pr ( k ) ∼ k − γ with γ = 3. Complex Networks Nutshell ◮ Typical for real networks: 2 < γ < 3. Basic models of complex networks ◮ Range true more generally for events with size Generalized random networks distributions that have power-law tails. Scale-free networks Small-world networks ◮ 2 < γ < 3: finite mean and ‘infinite’ variance (wild) Generalized affiliation networks ◮ In practice, γ < 3 means variance is governed by References upper cutoff. ◮ γ > 3: finite mean and variance (mild) 61 of 128
Overview of Examples Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks Nutshell WWW γ ≃ 2 . 1 for in-degree Basic models of WWW γ ≃ 2 . 45 for out-degree complex networks Generalized random Movie actors γ ≃ 2 . 3 networks Scale-free networks γ ≃ 2 . 8 Words (synonyms) Small-world networks Generalized affiliation networks References The Internets is a different business... 62 of 128
Overview of Real data Complex Networks Basic definitions Examples of From Barabási and Albert’s original paper [4] : Complex Networks Properties of Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References Fig. 1. The distribution function of connectivities for various large networks. ( A ) Actor collaboration graph with N � 212,250 vertices and average connectivity � k � � 28.78. ( B ) WWW, N � 325,729, � k � � 5.46 ( 6 ). ( C ) Power grid data, N � 4941, � k � � 2.67. The dashed lines have slopes (A) � actor � 2.3, (B) � www � 2.1 and (C) � power � 4. 63 of 128
Overview of Things to do and questions Complex Networks Basic definitions ◮ Vary attachment kernel. Examples of Complex Networks ◮ Vary mechanisms: Properties of Complex Networks 1. Add edge deletion Nutshell 2. Add node deletion Basic models of 3. Add edge rewiring complex networks ◮ Deal with directed versus undirected networks. Generalized random networks Scale-free networks ◮ Important Q.: Are there distinct universality classes Small-world networks Generalized affiliation networks for these networks? References ◮ Q.: How does changing the model affect γ ? ◮ Q.: Do we need preferential attachment and growth? ◮ Q.: Do model details matter? ◮ The answer is (surprisingly) yes. More later re Zipf. 64 of 128
Overview of Preferential attachment Complex Networks Basic definitions Examples of Complex Networks ◮ Let’s look at preferential attachment (PA) a little more Properties of Complex Networks closely. Nutshell ◮ PA implies arriving nodes have complete knowledge Basic models of complex networks of the existing network’s degree distribution. Generalized random networks ◮ For example: If P attach ( k ) ∝ k , we need to determine Scale-free networks Small-world networks the constant of proportionality. Generalized affiliation networks ◮ We need to know what everyone’s degree is... References ◮ PA is ∴ an outrageous assumption of node capability. ◮ But a very simple mechanism saves the day. . . 65 of 128
Overview of Preferential attachment through randomness Complex Networks Basic definitions Examples of ◮ Instead of attaching preferentially, allow new nodes Complex Networks to attach randomly. Properties of Complex Networks ◮ Now add an extra step: new nodes then connect to Nutshell some of their friends’ friends. Basic models of complex networks ◮ Can also do this at random. Generalized random networks ◮ Assuming the existing network is random, we know Scale-free networks Small-world networks Generalized affiliation probability of a random friend having degree k is networks References Q k ∝ kP k ◮ So rich-gets-richer scheme can now be seen to work in a natural way. 66 of 128
Overview of Robustness Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks Nutshell Basic models of ◮ System robustness and system robustness. complex networks Generalized random ◮ Albert et al., Nature, 2000: networks Scale-free networks “Error and attack tolerance of complex networks” [3] Small-world networks Generalized affiliation networks References 67 of 128
Overview of Robustness Complex Networks ◮ Standard random networks (Erd˝ os-Rényi) Basic definitions versus Examples of Complex Networks Scale-free networks Properties of Complex Networks b a Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References Scale-free Exponential from Albert et al., 2000 68 of 128
Overview of Robustness Complex Networks Basic definitions 12 a Examples of E SF Failure Complex Networks ◮ Plots of network 10 Attack Properties of diameter as a function Complex Networks 8 of fraction of nodes Nutshell 6 Basic models of removed complex networks 4 ◮ Erd˝ Generalized random 0.00 0.02 0.04 os-Rényi versus networks d Scale-free networks scale-free networks b c Small-world networks Generalized affiliation 15 ◮ blue symbols = networks Internet WWW 20 References random removal 10 Attack Attack 15 ◮ red symbols = 5 targeted removal Failure Failure 0 10 0.00 0.01 0.02 0.00 0.01 0.02 (most connected first) from f Albert et al., 2000 69 of 128
Overview of Robustness Complex Networks Basic definitions Examples of ◮ Scale-free networks are thus robust to random Complex Networks Properties of failures yet fragile to targeted ones. Complex Networks ◮ All very reasonable: Hubs are a big deal. Nutshell ◮ But: next issue is whether hubs are vulnerable or not. Basic models of complex networks Generalized random ◮ Representing all webpages as the same size node is networks Scale-free networks obviously a stretch (e.g., google vs. a random Small-world networks Generalized affiliation person’s webpage) networks References ◮ Most connected nodes are either: 1. Physically larger nodes that may be harder to ‘target’ 2. or subnetworks of smaller, normal-sized nodes. ◮ Need to explore cost of various targeting schemes. 70 of 128
Overview of People thinking about people: Complex Networks How are social networks structured? Basic definitions ◮ How do we define and measure connections? Examples of Complex Networks ◮ Methods/issues of self-report and remote sensing. Properties of Complex Networks Nutshell What about the dynamics of social networks? Basic models of complex networks ◮ How do social networks/movements begin & evolve? Generalized random networks Scale-free networks ◮ How does collective problem solving work? Small-world networks Generalized affiliation networks ◮ How does information move through social networks? References ◮ Which rules give the best ‘game of society?’ Sociotechnical phenomena and algorithms: ◮ What can people and computers do together? (google) ◮ Use Play + Crunch to solve problems. Which problems? 72 of 128
Overview of Social Search Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks A small slice of the pie: Nutshell Basic models of ◮ Q. Can people pass messages between distant complex networks Generalized random networks individuals using only their existing social Scale-free networks Small-world networks connections? Generalized affiliation networks ◮ A. Apparently yes... References 73 of 128
Overview of Milgram’s social search experiment (1960s) Complex Networks Basic definitions ◮ Target person = Examples of Complex Networks Boston stockbroker. Properties of Complex Networks ◮ 296 senders from Boston and Nutshell Omaha. Basic models of ◮ 20% of senders reached complex networks Generalized random target. networks Scale-free networks Small-world networks ◮ chain length ≃ 6.5. Generalized affiliation networks References Popular terms: ◮ The Small World Phenomenon; ◮ “Six Degrees of Separation.” http://www.stanleymilgram.com 74 of 128
Overview of The problem Complex Networks Basic definitions Lengths of successful chains: Examples of Complex Networks 18 Properties of Complex Networks 15 Nutshell From Travers and Basic models of 12 complex networks Milgram (1969) in Generalized random networks Sociometry: [26] Scale-free networks n ( L ) 9 Small-world networks “An Experimental Generalized affiliation networks Study of the Small 6 References World Problem.” 3 0 1 2 3 4 5 6 7 8 9 10 11 12 L 75 of 128
Overview of The problem Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks Nutshell Two features characterize a social ‘Small World’: Basic models of complex networks 1. Short paths exist Generalized random networks and Scale-free networks Small-world networks Generalized affiliation 2. People are good at finding them. networks References 76 of 128
Overview of Social Search Complex Networks Milgram’s small world experiment with email: Basic definitions Examples of Complex Networks Properties of Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References “An Experimental study of Search in Global Social Networks” P . S. Dodds, R. Muhamad, and D. J. Watts, Science , Vol. 301, pp. 827–829, 2003. [11] 77 of 128
Overview of Social search—the Columbia experiment Complex Networks Basic definitions Examples of Complex Networks Properties of ◮ 60,000+ participants in 166 countries Complex Networks ◮ 18 targets in 13 countries including Nutshell ◮ a professor at an Ivy League university, Basic models of complex networks ◮ an archival inspector in Estonia, Generalized random networks ◮ a technology consultant in India, Scale-free networks Small-world networks ◮ a policeman in Australia, Generalized affiliation networks and References ◮ a veterinarian in the Norwegian army. ◮ 24,000+ chains 78 of 128
Overview of Social search—the Columbia experiment Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks ◮ Milgram’s participation rate was roughly 75% Nutshell ◮ Email version: Approximately 37% participation rate. Basic models of complex networks ◮ Probability of a chain of length 10 getting through: Generalized random networks Scale-free networks . 37 10 ≃ 5 × 10 − 5 Small-world networks Generalized affiliation networks References ◮ ⇒ 384 completed chains (1.6% of all chains). 79 of 128
Overview of Social search—the Columbia experiment Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks ◮ Motivation/Incentives/Perception matter. Nutshell ◮ If target seems reachable Basic models of complex networks ⇒ participation more likely. Generalized random networks ◮ Small changes in attrition rates Scale-free networks Small-world networks ⇒ large changes in completion rates Generalized affiliation networks ◮ e.g., ց 15% in attrition rate References ⇒ ր 800% in completion rate 80 of 128
Overview of Social search—the Columbia experiment Complex Networks Basic definitions Successful chains disproportionately used Examples of Complex Networks ◮ weak ties (Granovetter) Properties of Complex Networks ◮ professional ties (34% vs. 13%) Nutshell ◮ ties originating at work/college Basic models of complex networks ◮ target’s work (65% vs. 40%) Generalized random networks Scale-free networks Small-world networks Generalized affiliation . . . and disproportionately avoided networks References ◮ hubs (8% vs. 1%) (+ no evidence of funnels) ◮ family/friendship ties (60% vs. 83%) Geography → Work 81 of 128
Overview of Social search—the Columbia experiment Complex Networks Basic definitions Examples of Complex Networks Senders of successful messages showed Properties of little absolute dependency on Complex Networks ◮ age, gender Nutshell Basic models of ◮ country of residence complex networks Generalized random ◮ income networks Scale-free networks Small-world networks ◮ religion Generalized affiliation networks ◮ relationship to recipient References Range of completion rates for subpopulations: 30% to 40% 82 of 128
Overview of Social search—the Columbia experiment Complex Networks Basic definitions Examples of Nevertheless, some weak discrepencies do exist... Complex Networks Properties of An above average connector: Complex Networks Nutshell Norwegian, secular male, aged 30-39, earning over Basic models of $100K, with graduate level education working in mass complex networks Generalized random media or science, who uses relatively weak ties to people networks Scale-free networks they met in college or at work. Small-world networks Generalized affiliation networks A below average connector: References Italian, Islamic or Christian female earning less than $2K, with elementary school education and retired, who uses strong ties to family members. 83 of 128
Overview of Social search—the Columbia experiment Complex Networks Basic definitions Examples of Complex Networks Mildly bad for continuing chain: Properties of Complex Networks choosing recipients because “they have lots of friends” or Nutshell because they will “likely continue the chain.” Basic models of complex networks Generalized random Why: networks Scale-free networks Small-world networks ◮ Specificity important Generalized affiliation networks ◮ Successful links used relevant information. References (e.g. connecting to someone who shares same profession as target.) 84 of 128
Overview of Social search—the Columbia experiment Complex Networks Basic definitions Examples of Complex Networks Basic results: Properties of Complex Networks ◮ � L � = 4 . 05 for all completed chains Nutshell Basic models of ◮ L ∗ = Estimated ‘true’ median chain length (zero complex networks attrition) Generalized random networks Scale-free networks ◮ Intra-country chains: L ∗ = 5 Small-world networks Generalized affiliation networks ◮ Inter-country chains: L ∗ = 7 References ◮ All chains: L ∗ = 7 ◮ Milgram: L ∗ ≃ 9 85 of 128
Overview of Usefulness: Complex Networks Basic definitions Harnessing social search: Examples of Complex Networks ◮ Can distributed social search be used for something Properties of Complex Networks big/good? Nutshell ◮ What about something evil? (Good idea to check.) Basic models of complex networks ◮ What about socio-inspired algorithms for information Generalized random networks search? (More later.) Scale-free networks Small-world networks ◮ For real social search, we have an incentives Generalized affiliation networks problem. References ◮ Which kind of influence mechanisms/algorithms would help propagate search? ◮ Fun, money, prestige, ... ? ◮ Must be ‘non-gameable.’ 86 of 128
Overview of Red balloons: Complex Networks A Grand Challenge: Basic definitions Examples of Complex Networks ◮ 1969: The Internet is born ( ⊞ ) Properties of (the ARPANET ( ⊞ )—four nodes!). Complex Networks ◮ Originally funded by DARPA who created a grand Nutshell Basic models of Network Challenge ( ⊞ ) for the 40th anniversary. complex networks Generalized random ◮ Saturday December 5, 2009: DARPA puts 10 red networks Scale-free networks weather balloons up during the day. Small-world networks Generalized affiliation networks ◮ Each 8 foot diameter balloon is anchored to the References ground somewhere in the United States. ◮ Challenge: Find the latitude and longitude of each balloon. ◮ Prize: $40,000. ∗ DARPA = Defense Advanced Research Projects Agency ( ⊞ ). 87 of 128
Overview of Where the balloons were: Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References 88 of 128
Overview of Finding red balloons: Complex Networks Basic definitions The winning team and strategy: Examples of Complex Networks ◮ MIT’s Media Lab ( ⊞ ) won in less that 9 hours. Properties of Complex Networks ◮ People were virally recruited online to help out. Nutshell ◮ Idea: Want people to both (1) find the balloons and Basic models of complex networks (2) involve more people. Generalized random networks ◮ Recursive incentive structure with exponentially Scale-free networks Small-world networks Generalized affiliation decaying payout: networks ◮ $2000 for correctly reporting the coordinates of a References balloon. ◮ $1000 for recruiting a person who finds a balloon. ◮ $500 for recruiting a person who recruits the balloon finder. ◮ etc. 89 of 128
Overview of Finding balloons: Complex Networks Clever scheme: Basic definitions ◮ Max payout = $4000 per balloon. Examples of Complex Networks ◮ Individuals have clear incentives to both Properties of 1. involve/source more people (spread), and Complex Networks 2. find balloons (goal action). Nutshell Basic models of ◮ Gameable? complex networks Generalized random ◮ Limit to how much money a set of bad actors can networks Scale-free networks extract. Small-world networks Generalized affiliation networks References Extra notes: ◮ MIT’s brand helped greatly. ◮ MIT group first heard about the competition a few days before. Ouch. ◮ A number of other teams did well ( ⊞ ). ◮ Worthwhile looking at these competing strategies. 90 of 128
Overview of The social world appears to be small... why? Complex Networks Basic definitions Examples of Complex Networks Theory: how do we understand the small world Properties of Complex Networks property? Nutshell ◮ Connected random networks have short average Basic models of complex networks path lengths: Generalized random networks � d AB � ∼ log ( N ) Scale-free networks Small-world networks Generalized affiliation networks N = population size, References d AB = distance between nodes A and B . ◮ But: social networks aren’t random... 91 of 128
Overview of Simple socialness in a network: Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks Nutshell Basic models of complex networks Need “clustering” (your Generalized random networks friends are likely to Scale-free networks Small-world networks know each other): Generalized affiliation networks References 92 of 128
Overview of Non-randomness gives clustering: Complex Networks Basic definitions B Examples of Complex Networks Properties of Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References A d AB = 10 → too many long paths. 93 of 128
Overview of Randomness + regularity Complex Networks Basic definitions B Examples of Complex Networks Properties of Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References A Now have d AB = 3 � d � decreases overall 94 of 128
Overview of Small-world networks Complex Networks Basic definitions Introduced by Watts and Strogatz (Nature, 1998) [30] Examples of “Collective dynamics of ‘small-world’ networks.” Complex Networks Properties of Small-world networks were found everywhere: Complex Networks Nutshell ◮ neural network of C. elegans, Basic models of complex networks ◮ semantic networks of languages, Generalized random networks ◮ actor collaboration graph, Scale-free networks Small-world networks Generalized affiliation ◮ food webs, networks References ◮ social networks of comic book characters,... Very weak requirements: ◮ local regularity + random short cuts 95 of 128
Overview of Toy model: Complex Networks Basic definitions Examples of Complex Networks Properties of Regular Small-world Random Complex Networks Nutshell Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References p = 0 p = 1 Increasing randomness 96 of 128
Overview of The structural small-world property: Complex Networks 1 Basic definitions Examples of C ( p ) / C (0) 0.8 Complex Networks Properties of Complex Networks 0.6 Nutshell Basic models of complex networks 0.4 Generalized random networks L ( p ) / L (0) Scale-free networks Small-world networks 0.2 Generalized affiliation networks References 0 0.0001 0.001 0.01 0.1 1 p ◮ L ( p ) = average shortest path length as a function of p ◮ C ( p ) = average clustring as a function of p 97 of 128
Overview of Previous work—finding short paths Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks But are these short cuts findable? Nutshell Basic models of Nope. complex networks Generalized random networks Nodes cannot find each other quickly Scale-free networks Small-world networks with any local search method. Generalized affiliation networks References Need a more sophisticated model... 98 of 128
Overview of Previous work—finding short paths Complex Networks Basic definitions ◮ What can a local search method reasonably use? Examples of Complex Networks ◮ How to find things without a map? Properties of Complex Networks ◮ Need some measure of distance between friends Nutshell and the target. Basic models of complex networks Generalized random networks Scale-free networks Some possible knowledge: Small-world networks Generalized affiliation networks ◮ Target’s identity References ◮ Friends’ popularity ◮ Friends’ identities ◮ Where message has been 99 of 128
Overview of Previous work—finding short paths Complex Networks Basic definitions Examples of Complex Networks Properties of Jon Kleinberg (Nature, 2000) [16] Complex Networks “Navigation in a small world.” Nutshell Basic models of complex networks Allowed to vary: Generalized random networks Scale-free networks Small-world networks 1. local search algorithm Generalized affiliation networks and References 2. network structure. 100 of 128
Overview of Previous work—finding short paths Complex Networks Kleinberg’s Network: Basic definitions Examples of Complex Networks 1. Start with regular d-dimensional cubic lattice. Properties of 2. Add local links so nodes know all nodes within a Complex Networks distance q . Nutshell Basic models of 3. Add m short cuts per node. complex networks Generalized random 4. Connect i to j with probability networks Scale-free networks Small-world networks Generalized affiliation − α . p ij ∝ x ij networks References ◮ α = 0: random connections. ◮ α large: reinforce local connections. ◮ α = d : connections grow logarithmically in space. 101 of 128
Overview of Previous work—finding short paths Complex Networks Basic definitions Examples of Complex Networks Theoretical optimal search: Properties of Complex Networks ◮ “Greedy” algorithm. Nutshell Basic models of ◮ Number of connections grow logarithmically (slowly) complex networks Generalized random in space: α = d . networks Scale-free networks ◮ Social golf. Small-world networks Generalized affiliation networks References Search time grows slowly with system size (like log 2 N ). But: social networks aren’t lattices plus links. 102 of 128
Overview of Previous work—finding short paths Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks ◮ If networks have hubs can also search well: Adamic Nutshell et al. (2001) [1] Basic models of P ( k i ) ∝ k − γ complex networks i Generalized random networks where k = degree of node i (number of friends). Scale-free networks Small-world networks ◮ Basic idea: get to hubs first Generalized affiliation networks (airline networks). References ◮ But: hubs in social networks are limited. 103 of 128
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