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. Frame 15/122
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 Frame 16/122
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.) Frame 17/122
Overview of Examples Complex Networks Basic definitions Examples of Complex Networks Properties of So what passes for a complex network? Complex Networks Nutshell ◮ Complex networks are large (in node number) Basic models of complex networks ◮ Complex networks are sparse (low edge to node Generalized random networks ratio) Scale-free networks Small-world networks ◮ Complex networks are usually dynamic and evolving Generalized affiliation networks ◮ Complex networks can be social, economic, natural, References informational, abstract, ... Frame 18/122
Overview of Examples Complex Networks Basic definitions Physical networks Examples of Complex Networks ◮ River networks ◮ The Internet Properties of ◮ Neural networks Complex Networks ◮ Road networks Nutshell ◮ Trees and leaves ◮ Power grids Basic models of ◮ Blood networks complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References ◮ Distribution (branching) versus redistribution (cyclical) Frame 19/122
Overview of Examples Complex Networks Interaction networks Basic definitions Examples of ◮ The Blogosphere Complex Networks Properties of ◮ Biochemical Complex Networks networks Nutshell ◮ Gene-protein Basic models of complex networks networks Generalized random networks Scale-free networks ◮ Food webs: who Small-world networks Generalized affiliation eats whom networks References ◮ The World Wide Web (?) ◮ Airline networks ◮ Call networks datamining.typepad.com ( ⊞ ) (AT&T) ◮ The Media Frame 20/122
Overview of Examples Complex Networks Basic definitions Interaction networks: Examples of social networks Complex Networks Properties of ◮ Snogging Complex Networks Nutshell ◮ Friendships Basic models of ◮ Acquaintances complex networks Generalized random networks ◮ Boards and Scale-free networks Small-world networks directors Generalized affiliation networks ◮ Organizations References ◮ facebook ( ⊞ ) twitter ( ⊞ ), (Bearman et al. , 2004) ◮ ‘Remotely sensed’ by: email activity, instant messaging, phone logs (*cough*). Frame 21/122
Overview of Examples 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 Frame 22/122
Overview of Examples Complex Networks Relational networks Basic definitions ◮ Consumer purchases Examples of (Wal-Mart: ≈ 1 petabyte = 10 15 bytes) Complex Networks Properties of ◮ Thesauri: Networks of words generated by meanings Complex Networks Nutshell ◮ Knowledge/Databases/Ideas Basic models of ◮ Metadata—Tagging: del.icio.us ( ⊞ ) flickr ( ⊞ ) complex networks Generalized random networks Scale-free networks Small-world networks common tags cloud | list Generalized affiliation networks community daily dictionary education encyclopedia References english free imported info information internet knowledge news reference research learning resource web2.0 wiki resources search tools useful web wikipedia Frame 23/122
Overview of Clickworthy Science: 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 Bollen et al. [5] Frame 24/122
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 Complex Networks ⇐ Typical hairball Nutshell ◮ number of nodes N = 500 Basic models of complex networks ◮ number of edges m = 1000 Generalized random networks Scale-free networks ◮ average degree � k � = 4 Small-world networks 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. Frame 25/122
Overview of Properties Complex Networks Basic definitions Examples of Some key features of real complex networks: Complex Networks Properties of ◮ Degree ◮ Concurrency Complex Networks distribution Nutshell ◮ Hierarchical Basic models of ◮ Assortativity scaling complex networks Generalized random ◮ Homophily ◮ Network distances networks Scale-free networks Small-world networks ◮ Clustering ◮ Centrality Generalized affiliation networks ◮ Motifs ◮ Efficiency References ◮ Modularity ◮ Robustness ◮ Coevolution of network structure and processes on networks. Frame 26/122
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks 1. Degree distribution P k Properties of Complex Networks ◮ P k is the probability that a randomly selected node Nutshell has degree k Basic models of complex networks ◮ Big deal: Form of P k key to network’s behavior Generalized random networks ◮ ex 1: Erdös-Rényi random networks have a Poisson Scale-free networks Small-world networks Generalized affiliation distribution: networks P k = e −� k � � k � k / k ! References ◮ ex 2: “Scale-free” networks: P k ∝ k − γ ⇒ ‘hubs’ ◮ We’ll come back to this business soon... Frame 27/122
Overview of Properties Complex Networks Basic definitions 2. Assortativity/3. Homophily: Examples of Complex Networks Properties of ◮ Social networks: Homophily ( ⊞ ) = birds of a feather Complex Networks ◮ e.g., degree is standard property for sorting: Nutshell Basic models of measure degree-degree correlations. complex networks ◮ Assortative network: [18] similar degree nodes Generalized random networks Scale-free networks connecting to each other. Small-world networks Generalized affiliation networks ◮ Often social: company directors, coauthors, actors. References ◮ Disassortative network: high degree nodes connecting to low degree nodes. ◮ Often techological or biological: Internet, protein interactions, neural networks, food webs. Frame 28/122
Overview of Properties Complex Networks Basic definitions 4. Clustering: Examples of Complex Networks ◮ Your friends tend to know each other. Properties of Complex Networks ◮ Two measures: Nutshell Basic models of �� � j 1 j 2 ∈N i a j 1 j 2 complex networks due to Watts & Strogatz [28] C 1 = Generalized random k i ( k i − 1 ) / 2 networks i Scale-free networks Small-world networks Generalized affiliation C 2 = 3 × # triangles networks due to Newman [19] References # 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. Frame 29/122
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks Properties of 5. Motifs: Complex Networks Nutshell ◮ Small, recurring functional subnetworks Basic models of ◮ e.g., Feed Forward Loop: complex networks Generalized random networks a feedforward loop Scale-free networks X Small-world networks X Generalized affiliation networks Y Y References n Z Shen-Orr, Uri Alon, et al. [21] Frame 30/122
Overview of Properties Complex Networks 6. modularity: Basic definitions Examples of Complex Networks 49 53 58 Properties of 63 46 83 114 Complex Networks 28 33 11 25 97 88 1 59 67 Nutshell 73 105 24 50 103 37 Basic models of 89 69 36 45 110 109 complex networks 57 90 44 66 34 42 Generalized random 16 75 82 4 networks 31 93 86 91 112 80 Scale-free networks 0 48 18 54 9 92 Small-world networks 23 7 29 Generalized affiliation 104 8 61 71 94 networks 41 35 78 68 99 22 19 References 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 [8] : NCAA football Frame 31/122
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks Properties of 7. Concurrency: Complex Networks Nutshell ◮ Transmission of a contagious element only occurs Basic models of during contact [16] complex networks Generalized random networks ◮ Rather obvious but easily missed in a simple model Scale-free networks Small-world networks ◮ Dynamic property—static networks are not enough Generalized affiliation networks ◮ Knowledge of previous contacts crucial References ◮ Beware cumulated network data! Frame 32/122
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 Complex Networks ◮ Reveals fractal nature of natural branching networks ◮ Hierarchy is not pure but mixed (Tokunaga). [23, 10] Nutshell ◮ Major examples: rivers and blood networks. Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References (c) (b) (a) ◮ Beautifully described but poorly explained. Frame 33/122
Overview of Properties Complex Networks Basic definitions Examples of 9. Network distances: Complex Networks Properties of (a) shortest path length d ij : Complex Networks Nutshell ◮ Fewest number of steps between nodes i and j . Basic models of complex networks ◮ (Also called the chemical distance between i and j .) Generalized random networks Scale-free networks Small-world networks Generalized affiliation (b) average path length � d ij � : networks References ◮ Average shortest path length in whole network. ◮ Good algorithms exist for calculation. ◮ Weighted links can be accommodated. Frame 34/122
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks 9. Network distances: Properties of Complex Networks (c) Network diameter d max : Nutshell ◮ Maximum shortest path length in network. Basic models of complex networks Generalized random networks Scale-free networks ij d − 1 � n � ] − 1 : (d) Closeness d cl = [ � / Small-world networks ij 2 Generalized affiliation networks ◮ Average ‘distance’ between any two nodes. References ◮ Closeness handles disconnected networks ( d ij = ∞ ) ◮ d cl = ∞ only when all nodes are isolated. Frame 35/122
Overview of Properties Complex Networks Basic definitions Examples of Complex Networks 10. Centrality: Properties of Complex Networks ◮ Many such measures of a node’s ‘importance.’ Nutshell ◮ ex 1: Degree centrality: k i . Basic models of complex networks ◮ ex 2: Node i ’s betweenness Generalized random networks Scale-free networks = fraction of shortest paths that pass through i . Small-world networks Generalized affiliation ◮ ex 3: Edge ℓ ’s betweenness networks References = fraction of shortest paths that travel along ℓ . ◮ ex 4: Recursive centrality: Hubs and Authorities (Jon Kleinberg [15] ) Frame 36/122
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). Frame 37/122
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... Frame 38/122
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 complex networks 2. scale-free networks Generalized random networks Scale-free networks 3. small-world networks Small-world networks Generalized affiliation networks 4. statistical generative models ( p ∗ ) References 5. generalized affiliation networks Frame 39/122
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 from P k . Scale-free networks Small-world networks ◮ Wire nodes together randomly. Generalized affiliation networks ◮ Create ensemble to test deviations from References randomness. Frame 41/122
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. Frame 42/122
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 (A) (B) Small-world networks 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. Frame 43/122
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 i 4 e networks i 3 2 Scale-free networks Small-world networks i 2 Generalized affiliation networks i 1 ◮ Rewire one end of each edge. References ◮ 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 Frame 44/122
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 Phase 3: Small-world networks Generalized affiliation networks ◮ Randomize network wiring by applying rewiring References algorithm liberally. ◮ Rule of thumb: # Rewirings ≃ 10 × # edges [17] . Frame 45/122
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” [3] ◮ Somewhat misleading nomenclature... Frame 47/122
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. . . Frame 48/122
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 Frame 49/122
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? Frame 50/122
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. Frame 51/122
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 . Frame 52/122
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 . Frame 53/122
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 . Frame 54/122 ◮ Next find c i . . .
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). Frame 55/122
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 Frame 56/122
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 Frame 57/122
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 Frame 58/122
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) Frame 59/122
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... Frame 60/122
Overview of Real data Complex Networks Basic definitions Examples of From Barabási and Albert’s original paper [3] : 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. Frame 61/122
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. Frame 62/122
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. . . Frame 63/122
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. Frame 64/122
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” [2] Small-world networks Generalized affiliation networks References Frame 65/122
Overview of Robustness Complex Networks ◮ Standard random networks (Erdös-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 Frame 66/122 Albert et al., 2000
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 ◮ Erdös-Rényi versus Generalized random 0.00 0.02 0.04 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 Frame 67/122
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. Frame 68/122
Some problems for people thinking about Overview of Complex Networks people?: Basic definitions Examples of Complex Networks How are social networks structured? Properties of ◮ How do we define connections? Complex Networks Nutshell ◮ How do we measure connections? Basic models of complex networks ◮ (remote sensing, self-reporting) Generalized random networks Scale-free networks Small-world networks What about the dynamics of social networks? Generalized affiliation networks References ◮ How do social networks evolve? ◮ How do social movements begin? ◮ How does collective problem solving work? ◮ How is information transmitted through social networks? Frame 70/122
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 Frame 71/122
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 Frame 72/122
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: [24] 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 Frame 73/122
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 Frame 74/122
Overview of Social Search Complex Networks Milgram’s small world experiment with e-mail [9] 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 Frame 75/122
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 Frame 76/122
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). Frame 77/122
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 Frame 78/122
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 Frame 79/122
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% Frame 80/122
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. Frame 81/122
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.) Frame 82/122
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 Frame 83/122
Overview of The social world appears to be small... Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks ◮ Connected random networks have short average Nutshell path lengths: Basic models of complex networks � d AB � ∼ log ( N ) Generalized random networks Scale-free networks N = population size, Small-world networks Generalized affiliation networks d AB = distance between nodes A and B . References ◮ But: social networks aren’t random... Frame 84/122
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 Frame 85/122
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. Frame 86/122
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 Frame 87/122
Overview of Small-world networks Complex Networks Basic definitions Introduced by Watts and Strogatz (Nature, 1998) [28] 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 Frame 88/122
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 Frame 89/122
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 Frame 90/122
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... Frame 91/122
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 Frame 92/122
Overview of Previous work—finding short paths Complex Networks Basic definitions Examples of Complex Networks Properties of Jon Kleinberg (Nature, 2000) [14] 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. Frame 93/122
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 Nutshell distance q . Basic models of 3. Add m short cuts per node. complex networks Generalized random networks 4. Connect i to j with probability 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 : same number of connections at all scales. Frame 94/122
Overview of Previous work—finding short paths Complex Networks Basic definitions Examples of Complex Networks Properties of Complex Networks Theoretical optimal search: Nutshell ◮ “Greedy” algorithm. Basic models of complex networks ◮ Same number of connections at all scales: α = d . Generalized random networks Scale-free networks Small-world networks Generalized affiliation Search time grows slowly with system size (like log 2 N ). networks References But: social networks aren’t lattices plus links. Frame 95/122
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. Frame 96/122
Overview of The problem Complex Networks Basic definitions Examples of Complex Networks Properties of If there are no hubs and no underlying lattice, how can Complex Networks search be efficient? Nutshell Basic models of Which friend of a is closest complex networks Generalized random b to the target b? networks Scale-free networks Small-world networks Generalized affiliation networks What does ‘closest’ mean? References a What is ‘social distance’? Frame 98/122
Overview of Models Complex Networks Basic definitions Examples of One approach: incorporate identity. Complex Networks Properties of Complex Networks Identity is formed from attributes such as: Nutshell ◮ Geographic location Basic models of ◮ Type of employment complex networks Generalized random networks ◮ Religious beliefs Scale-free networks Small-world networks ◮ Recreational activities. Generalized affiliation networks References Groups are formed by people with at least one similar attribute. Attributes ⇔ Contexts ⇔ Interactions ⇔ Networks. Frame 99/122
Overview of Social distance—Bipartite affiliation networks Complex Networks contexts 1 2 3 4 Basic definitions Examples of Complex Networks Properties of Complex Networks Nutshell Basic models of individuals complex networks a b c d e Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References b d unipartite a network e c Bipartite affiliation networks: boards and directors, Frame 100/122 movies and actors.
Overview of Social distance—Context distance Complex Networks Basic definitions occupation Examples of Complex Networks Properties of Complex Networks education health care Nutshell Basic models of complex networks Generalized random kindergarten high school networks doctor teacher teacher nurse Scale-free networks Small-world networks Generalized affiliation networks References a b c d e Frame 101/122
Overview of Models Complex Networks Basic definitions Examples of Complex Networks Distance between two individuals x ij is the height of Properties of Complex Networks lowest common ancestor. Nutshell l=4 Basic models of complex networks b=2 Generalized random networks Scale-free networks Small-world networks g=6 Generalized affiliation networks j References v i k x ij = 3, x ik = 1, x iv = 4. Frame 102/122
Overview of Models Complex Networks Basic definitions Examples of Complex Networks ◮ Individuals are more likely to know each other the Properties of Complex Networks closer they are within a hierarchy. Nutshell ◮ Construct z connections for each node using Basic models of complex networks Generalized random networks p ij = c exp {− α x ij } . Scale-free networks Small-world networks Generalized affiliation networks References ◮ α = 0: random connections. ◮ α large: local connections. Frame 103/122
Overview of Models Complex Networks Basic definitions Examples of Complex Networks Generalized affiliation networks Properties of geography occupation age Complex Networks Nutshell 0 100 Basic models of complex networks Generalized random networks Scale-free networks Small-world networks Generalized affiliation networks References a b c d e ◮ Blau & Schwartz [4] , Simmel [22] , Breiger [7] , Watts et al. [27] Frame 104/122
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