Recommended
More Related Content
Similar to MSCX2023_Sergio Gomez_PartI
Similar to MSCX2023_Sergio Gomez_PartI (20)
Recently uploaded
Recently uploaded (20)
MSCX2023_Sergio Gomez_PartI
- 1. Structure of complex networks Sergio Gómez Universitat Rovira i Virgili, Tarragona (Spain) MSCxNet 2023 Catania
- 5. Tarragona 5 About Tarraco, capital of the Hispania Tarraconensis during the Roman Empire
- 7. Tarragona 7 About Port Aventura + Ferrari Land
- 9. 9 Outline ◼ Complex networks? ◼ Representation of complex networks ◼ Structure of complex networks ◼ Models of complex networks Structure of complex networks
- 10. 10 Outline ◼ Complex networks? ◼ Representation of complex networks ◼ Structure of complex networks ◼ Models of complex networks Structure of complex networks
- 11. 11 Complex networks? ◼ Network (Graph) G = (V,E) Nodes (Vertices) ◼ Any kind of entity Links (Edges) ◼ Relationships or interactions between nodes Complex Networks
- 12. 12 Complex networks? ◼ Network (Graph) G = (V,E) Nodes (Vertices) ◼ Any kind of entity Links (Edges) ◼ Relationships or interactions between nodes Complex Networks Network = Representation for pair-wise relations
- 13. 13 Complex networks ◼ Why complex? Not regular Not completely random Complex Networks
- 14. 14 What are not Complex networks Complex Networks ◼ Complex ≠ Complicated
- 15. 15 ◼ Telephony 4G network (LTE) Complex Networks Examples of real complex networks
- 16. ◼ Internet Autonomous systems ◼ Nodes: 23000 autonomous systems (year 2006) ◼ Links: connections between them 16 Complex Networks
- 17. ◼ Transportation networks World air transportation network ◼ Nodes: 3618 airports ◼ Links: 27028 flight connections 17 Complex Networks
- 18. 18 Complex Networks ◼ Biological networks Escherichia coli metabolic network ◼ Nodes: 473 metabolites ◼ Links: 574 metabolic interactions
- 19. ◼ Biological networks Cortical networks ◼ Nodes: cortical areas ◼ Links: functional connectivity between them (EEG, FRMI, …) 19 Complex Networks
- 20. ◼ Ecological networks Little Rock Lake (Wisconsin) food web ◼ Nodes: 182 taxa ◼ Links: 2494 relationships of who predates on whom 20 Complex Networks
- 21. ◼ Social networks 15M social movement ◼ Nodes: 87,569 users ◼ Links: 581,749 Twitter messages 21 Complex Networks
- 22. ◼ Social networks Dolphin's network ◼ Nodes: 62 dolphins ◼ Links: 159 non-random social interactions 22 Complex Networks
- 23. ◼ Economy World trade network ◼ Nodes: countries ◼ Links: amount of trade between them 23 Complex Networks
- 24. ◼ Summary Complex networks everywhere! ◼ Technology ◼ Transportation ◼ Biology ◼ Ecology ◼ Chemistry ◼ Social sciences ◼ Economy ◼ Business ◼ Linguistics ◼ Politics ◼ … 24 Complex Networks
- 25. 25 ◼ Real networks have common characteristics Sparse Complex structure Complex dynamics on networks Complex dynamics of networks Complex Networks Characterizing real complex networks
- 26. 26 Sparse ◼ N: number of Nodes ◼ M: number of Edges ◼ M ∼ O(N) instead of O(N2) Complex Networks
- 27. 27 Sparse ◼ N: number of Nodes ◼ M: number of Edges ◼ M ∼ O(N) instead of O(N2) Examples ◼ Dolphin's network Nodes: 62 dolphins Links: 159 social interactions ≪ 1,891 possible interactions Average degree: 5.13 edges / node ◼ World air transportation network Nodes: 3,618 airports Links: 27,028 connections ≪ 6,543,153 possible connections Average degree: 14.94 edges / node Complex Networks
- 28. 28 Complex structure ◼ Power law degree distribution ◼ Small-world property ◼ Community structure ◼ High transitivity ◼ Assortativity ◼ Rich club ◼ Motifs Complex Networks
- 29. 29 Complex dynamics on networks ◼ Diffusion ◼ Epidemic spreading ◼ Rumor and opinion spreading ◼ Synchronization ◼ Random walks ◼ Routing ◼ Evolutionary games Complex Networks
- 30. 30 Complex dynamics of networks ◼ Evolving networks ◼ Percolation Complex Networks
- 31. 31 Outline ◼ Complex networks? ◼ Representation of complex networks ◼ Structure of complex networks ◼ Models of complex networks Structure of complex networks
- 32. 32 ◼ Mathematical representation ◼ Computational representation Representation Representation of complex networks
- 33. 33 Mathematical ◼ Adjacency matrix 𝐴 = 𝑎𝑖𝑗 ∈ ℝ𝑁×𝑁 , 𝑎𝑖𝑗 ∈ 0,1 , 𝑖, 𝑗 ∈ 1, … , 𝑁 𝐴 = 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 𝐴 = 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 𝐴 = 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 Representation
- 34. 34 Mathematical ◼ Adjacency matrix 𝐴 = 𝑎𝑖𝑗 ∈ ℝ𝑁×𝑁 , 𝑎𝑖𝑗 ∈ 0,1 , 𝑖, 𝑗 ∈ 1, … , 𝑁 ◼ Undirected network Symmetric adjacency matrix 𝐴 = 𝐴𝑇 ◼ Weighted network Weights matrix 𝑊 = 𝑤𝑖𝑗 ∈ ℝ𝑁×𝑁 , 𝑤𝑖𝑗 ∈ ℝ, 𝑖, 𝑗 ∈ 1, … , 𝑁 Representation
- 35. 35 Mathematical ◼ Degree 𝑘𝑖 of a node 𝑖 = number of adjacent edges 𝑘𝑖 = 𝑗=1 𝑁 𝑎𝑖𝑗 , 𝑖 ∈ 1, … , 𝑁 ◼ Total number of edges of an undirected network 𝐿 = 1 2 𝑖=1 𝑁 𝑘𝑖 = 1 2 𝑖=1 𝑁 𝑗=1 𝑁 𝑎𝑖𝑗 ◼ Average degree 𝑘 of an undirected network 𝑘 = 2𝐿 𝑁 Representation
- 36. 36 Mathematical ◼ Density 𝜌 of an undirected network 𝜌 = 2𝐿 𝑁 𝑁 − 1 ◼ Sparse networks 𝐿 ∼ 𝑂 𝑁 𝐿 = 𝑐𝑁 𝜌 = 2𝑐 𝑁 − 1 ≪ 1 Representation
- 37. 37 Computational ◼ Complex networks are sparse Avoid adjacency matrices! Adjacency matrix has 𝑁2 elements to store Sparse complex networks 𝑂(𝑁) non-zero elements Representation
- 38. 38 Computational ◼ Complex networks are sparse Avoid adjacency matrices! Adjacency matrix has 𝑁2 elements to store Sparse complex networks 𝑂(𝑁) non-zero elements ◼ Use Adjacency lists Lists of edges Sparse matrix structures (equivalent to adjacency lists) Representation
- 39. 39 Computational Representation 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 2 3 4 1 2 3 4 3 4 1 1 4 4 2 3 3 1 3 1 4 2 4 3 4 4 4 Adjacency matrix (𝑁 × 𝑁) List of edges (𝐿 × 2) Adjacency list (𝑁 + 2𝐿)
- 40. 40 Computational ◼ Use already-implemented graph or network libraries Julia: Graphs Python: igraph, networkx, graph-tool R: igraph C++: igraph, boost (BGL) Java: JGraphT Ada: radalib Representation
- 41. 41 Computational ◼ Implementation of algorithms Ensure you use the right type: graph, digraph, multigraph, etc. Use library functions to iterate over neighbors Avoid search of edges (implicit or explicit) Allocate memory for all variables before any loop Avoid converting iterators to arrays (collect, etc.) If algorithm is time consuming, use compiled languages (Julia, C++, Java, etc.) In python, always use numpy arrays, not lists Representation
- 42. 42 Outline ◼ Complex networks? ◼ Representation of complex networks ◼ Structure of complex networks ◼ Models of complex networks Structure of complex networks
- 43. 43 ◼ Type of network By type of edge By type of node By attributes ◼ Level of description Microscale Macroscale Mesoscale Structure Structure of complex networks
- 44. 44 Type of network ◼ By type of edge Undirected, Directed Multilayer: edges of different types Hypergraph: hyperedges (higher order relations) Simplicial complex: downward-closed hypergraph ◼ By type of node (and distribution of edges) Unipartite, Bipartite, Tripartite, etc. Network of networks ◼ By attributes Edges with real values: Weighted, unweighted, signed Annotated graphs: attributes in nodes, attributes in edges Structure
- 45. 45 Level of description ◼ Microscale: at the level of nodes ◼ Mesoscale: at the level of groups of nodes ◼ Macroscale: at the level of the whole network Structure Macroscale Mesoscale Microscale
- 46. 46 Microscale Structure Macroscale Mesoscale Microscale
- 47. 47 Microscale ◼ Basic node descriptors Degree 𝑘𝑖 (undirected networks) 𝑘𝑖 = 𝑗=1 𝑁 𝑎𝑖𝑗 Output degree 𝑘𝑖 out , input degree 𝑘𝑖 in (directed networks) 𝑘𝑖 out = 𝑗=1 𝑁 𝑎𝑖𝑗 , 𝑘𝑖 in = 𝑗=1 𝑁 𝑎𝑗𝑖 Strength 𝑠𝑖 (undirected networks) Output strength 𝑠𝑖 out , intput strength 𝑠𝑖 in (directed networks) Structure
- 48. 48 Microscale ◼ Clustering coefficient Measures the transitivity of the connections: fraction of neighbors that are connected between them 𝐶𝑖 = σ𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 𝑎𝑗𝑘 σ𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 = 2 𝑘𝑖 𝑘𝑖 − 1 𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 𝑎𝑗𝑘 = = 𝐴3 𝑖𝑖 𝐴𝐹𝐴 𝑖𝑖 , 𝐹𝑖𝑗 = 1 − 𝛿𝑖𝑗 Structure
- 49. 49 Microscale ◼ Clustering coefficient Measures the transitivity of the connections: fraction of neighbors that are connected between them 𝐶𝑖 = σ𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 𝑎𝑗𝑘 σ𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 = 2 𝑘𝑖 𝑘𝑖 − 1 𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 𝑎𝑗𝑘 = = 𝐴3 𝑖𝑖 𝐴𝐹𝐴 𝑖𝑖 , 𝐹𝑖𝑗 = 1 − 𝛿𝑖𝑗 Node 1 neighbors: 4 connected neighbors: 2 non-connected neighbors: 4 pairs of neighbors: 6 𝐶𝑖 = 2/6 = 0.33 Structure 1 2 3 4 5
- 50. 50 Microscale ◼ Clustering coefficient Measures the transitivity of the connections: fraction of neighbors that are connected between them 𝐶𝑖 = σ𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 𝑎𝑗𝑘 σ𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 = 2 𝑘𝑖 𝑘𝑖 − 1 𝑗<𝑘 𝑎𝑖𝑗 𝑎𝑖𝑘 𝑎𝑗𝑘 = = 𝐴3 𝑖𝑖 𝐴𝐹𝐴 𝑖𝑖 , 𝐹𝑖𝑗 = 1 − 𝛿𝑖𝑗 Never use these formulas for large networks! Structure
- 51. 51 Microscale ◼ Clustering coefficient Structure
- 52. 52 Microscale ◼ Centrality measures: importance of the nodes according to different criteria Number of neighbors (degree) Distances to the rest of the nodes Being in the middle of shortest paths Importance of one node depends on the importance of the neighbors Frequently, only the ranking of nodes is considered Structure S Gómez Centrality in networks: Finding the most important nodes Business and Consumer Analytics: New Ideas P. Moscato and N. J. de Vries (eds.), Springer (2019) 401-43
- 53. 53 Microscale ◼ Centrality measures Closeness: how close the node is to the rest 𝑑𝑖𝑗: distance between nodes 𝑖 and 𝑗 𝐶𝐿𝑖 = 1 σ𝑗 𝑑𝑖𝑗 , 𝐶𝐿𝑖 = 𝑁 σ𝑗 𝑑𝑖𝑗 , 𝐶𝐿𝑖 = 1 𝑁 − 1 𝑗≠𝑖 1 𝑑𝑖𝑗 Structure
- 54. 54 Microscale ◼ Centrality measures Betweenness: how frequently the node is in the middle of shortest paths 𝜎𝑠𝑑: number of shortest paths between nodes 𝑠 and 𝑑 𝜎𝑠𝑑 𝑖 : number of shortest paths between nodes 𝑠 and 𝑑 that cross node 𝑖 𝐵𝑖 = 1 𝑁 − 1 𝑁 − 2 𝑠,𝑑 𝑠≠𝑑≠𝑖 𝜎𝑠𝑑 𝑖 𝜎𝑠𝑑 Structure
- 55. 55 Microscale ◼ Centrality measures Eigenvector centrality: the importance of the node is proportional to the sum of the importance of the neighbors 𝐶𝑖 eig = 1 𝜆 𝑗 𝑎𝑗𝑖 𝐶𝑗 eig , 𝐴𝑪 eig = 𝜆𝑪 eig Structure
- 56. 56 Microscale ◼ Centrality measures Katz centrality: generalization of eigenvector centrality to account for not only the immediate neighbors 𝐶𝑖 kat𝑧 = 1 𝜆 𝑘=0 ∞ 𝑗 𝛼𝑘 𝐴𝑘 𝑗𝑖 , 𝑪 ka𝑡𝑧 = 𝛼𝐴𝑇 𝑪 ka𝑡𝑧 + 𝟏 Structure
- 57. 57 Microscale ◼ Centrality measures Hubs and authorities: for directed networks, separate the importance between sending and receiving links 𝑪 auth = 𝛼𝐴𝑇 𝑪 hub , 𝑪 hub = 𝛽𝐴𝑪 auth 𝐴𝑇 𝐴 𝑪 auth = 𝛾𝑪 auth , 𝐴𝐴𝑇 𝑪 hub = 𝛾𝑪 hub 𝛾 = 𝛼𝛽 −1 𝑪 auth = 𝐴𝑇 𝑪 ℎub , 𝑪 hub = 𝐴𝑪 autℎ Structure
- 58. 58 Microscale ◼ Centrality measures PageRank: Like eigenvector centrality, but normalizing the importance of the neighbor by the number of links it sends 𝐶𝑖 pr = 𝛼 𝑗 𝑎𝑗𝑖 𝐶𝑗 pr 𝑘𝑗 out + 1 − 𝛼 𝑁 , 𝛼 = 0.85 Can be interpreted as the occupation probability of a random walker, with a term of teleportation to scape from sinks Structure
- 59. 59 Microscale ◼ Centrality measures Structure Degree Closeness Betweenness
- 60. 60 Microscale ◼ Centrality measures Structure Eigenvector Katz PageRank
- 61. 61 Macroscale Structure Macroscale Mesoscale Microscale
- 62. 62 Macroscale ◼ Basic global descriptors Number of nodes 𝑁 Number of edges 𝐿 Average degree 𝑘 Density 𝜌 Average clustering coefficient 𝐶 Structure
- 63. 63 Macroscale ◼ Connectedness Undirected networks ▪ Connected components Directed networks ▪ Strongly connected components ▪ Weakly connected components ▪ In-component of a node ▪ Out-component of a node Structure
- 64. 64 Macroscale ◼ Degree distribution Most real networks have power-law degree distributions 𝑃 𝑘 ∝ 𝑘−𝛾, 2 < 𝛾 < 3 Structure
- 65. 65 Macroscale ◼ Degree distribution Most real networks have power-law degree distributions 𝑃 𝑘 = 𝑎 𝑘−𝛾, 2 < 𝛾 < 3 Structure Slope of log-log histogram −𝛾 + 1 log 𝑃 𝑘 = −𝛾 log 𝑘 + log 𝑎 In log-log scale
- 66. 66 Macroscale ◼ Small-world property In most real network, the distances are very small Average path length ℓ = 2 𝑁 𝑁 − 1 𝑖<𝑗 𝑑𝑖𝑗 Diameter 𝐷 = max 𝑖<𝑗 𝑑𝑖𝑗 Structure
- 67. 67 Macroscale ◼ Small-world property In most real network, the distances are very small Structure
- 68. 68 Macroscale ◼ Small-world property In most real network, the distances are very small Structure
- 69. 69 Macroscale ◼ Assortative mixing Assortative networks: high-degree nodes tend to connect to high-degree nodes Disassortative networks: high-degree nodes tend to connect to low-degree nodes Assortativity 𝑟 = σ𝑖,𝑗 𝑎𝑖𝑗 − 𝑘𝑖𝑘𝑗/2𝐿 𝑘𝑖𝑘𝑗 σ𝑖,𝑗 𝑘𝑖𝛿𝑖𝑗 − 𝑘𝑖𝑘𝑗/2𝐿 𝑘𝑖𝑘𝑗 Correlation between the degrees of nodes at both sides of every edge Structure
- 70. 70 Macroscale ◼ Assortative mixing Structure
- 71. 71 Mesoscale Structure Macroscale Mesoscale Microscale
- 72. 72 Mesoscale ◼ Community structure ◼ Rich-club ◼ Core-periphery ◼ Motifs ◼ Shells Structure
- 73. 73 Outline ◼ Complex networks? ◼ Representation of complex networks ◼ Structure of complex networks ◼ Models of complex networks Structure of complex networks
- 74. 74 Models of complex networks ◼ Models ◼ Regular networks ◼ Complex network models Models
- 75. 75 Models of complex networks ◼ Models For networks For processes on networks For processes of networks ◼ Why models? We have data of real networks! We know the details of real processes on real networks! We know the evolution of real networks! Models
- 76. 76 ◼ Why models? Because... Data does not imply knowledge! Models
- 77. 77 ◼ Why models? Because... Data does not imply knowledge! Models provide ◼ Explanation ◼ Prediction Models
- 78. 78 ◼ Why models? Because... Data does not imply knowledge! Models provide ◼ Explanation ◼ Prediction Models must be ◼ Simple (Occam’s razor) ◼ Accurate Models Ptolemy Tycho Brahe Kepler Copernicus
- 79. 79 ◼ Why models? Because... Data does not imply knowledge! Models provide ◼ Explanation ◼ Prediction Models must be ◼ Simple (Occam’s razor) ◼ Accurate Note: Agent-based models ◼ Not simple, but more realistic ◼ Sometimes difficult (even impossible) to identify relationships ◼ Usually cannot provide explanations, even if accurate and with good predictions Models
- 80. 80 ◼ Network models To explain the appearance of topological features ◼ Power-law degree distribution, small-world property, clustering, community structure, etc. Models
- 81. 81 ◼ Network models To explain the appearance of topological features ◼ Power-law degree distribution, small-world property, clustering, community structure, etc. To investigate the relationship between topology and function ◼ Which features are needed for a certain phenomenon? Models
- 82. 82 ◼ Network models To explain the appearance of topological features ◼ Power-law degree distribution, small-world property, clustering, community structure, etc. To investigate the relationship between topology and function ◼ Which features are needed for a certain phenomenon? To understand processes on networks ◼ Diffusion, epidemic spreading, rumors, innovations, collaboration, competition, evolutionary games, routing, congestion, synchronization, etc. Models
- 83. 83 ◼ Network models To explain the appearance of topological features ◼ Power-law degree distribution, small-world property, clustering, community structure, etc. To investigate the relationship between topology and function ◼ Which features are needed for a certain phenomenon? To understand processes on networks ◼ Diffusion, epidemic spreading, rumors, innovations, collaboration, competition, evolutionary games, routing, congestion, synchronization, etc. To understand processes of networks ◼ Network formation, evolution, interaction, etc. Models
- 84. 84 ◼ Example Models ◼ Network model SF ◼ Dynamics model Kuramoto
- 85. 85 ◼ Example Models ◼ Real network Star (proxy of SF) ◼ Real dynamics Rössler chaotic oscillator
- 86. 86 Regular networks ◼ Traditional networks used in physics and engineering ◼ Sometimes allow analytical solutions ◼ Discretization of continuous space Regular networks
- 87. 87 ◼ Toy networks Line Star Ring Fully connected Regular networks
- 89. 89 ◼ Lattices Without and with periodic boundary conditions Regular networks
- 90. 90 ◼ Other Bethe lattices / Cayley trees (internal nodes of same k) Ramanujan graphs (large spectral gap) Regular networks
- 91. 91 ◼ Other Fractal networks Regular networks
- 92. 92 Complex network models ◼ Models for these properties Degree distribution Average path length Clustering Communities Other Complex network models
- 93. 93 Models according to degree distribution ◼ Erdős-Rényi model (ER) ◼ Barabási-Albert model (BA) ◼ Configuration model (CM) Complex network models
- 94. 94 Models according to degree distribution ◼ Erdős-Rényi model (ER) ◼ Barabási-Albert model (BA) ◼ Configuration model (CM) Complex network models
- 95. 95 ◼ Erdős-Rényi model (ER) Model GN,K by Erdős & Rényi (1959) ◼ 𝑁: number of nodes ◼ 𝐾: number of edges (0 ≤ 𝐾 ≤ 𝑁(𝑁 − 1)/2) ◼ Each edge connects a randomly selected (and not previously connected) pair of nodes Model GN,p by Gilbert (1959) ◼ 𝑁: number of nodes ◼ 𝑝: probability of having an edge (0 ≤ 𝑝 ≤ 1) ◼ Each pair of nodes has a probability 𝑝 of having an edge Complex network models
- 96. 96 ◼ Erdős-Rényi model (ER) Model GN,p Complex network models p=0.1 p=0.2 p=0.3 p=0.0
- 97. 97 ◼ Erdős-Rényi model (ER) Relationship between GN,K and GN,p ◼ GN,p to GN,K ◼ GN,K to GN,p Property ◼ Almost surely, connected network if Complex network models
- 98. 98 ◼ Erdős-Rényi model (ER) Degree distribution ◼ Binomial ◼ Poisson, in the limit while constant Complex network models
- 99. 99 Models according to degree distribution ◼ Erdős-Rényi model (ER) ◼ Barabási-Albert model (BA) ◼ Configuration model (CM) Complex network models
- 100. 100 ◼ Barabási-Albert model (BA) Based on growth and preferential attachment (1999) ◼ 𝑁: number of nodes ◼ 𝑚0: number of initial nodes (𝑚0 ≤ 𝑁) ◼ 𝑚: number of edges for each new node (𝑚 ≤ 𝑚0) ◼ The network begins with an initial small connected network containing 𝑚0 nodes ◼ New nodes are added until the network has the desired 𝑁 nodes (growth) ◼ Each new node establishes 𝑚 edges to the current available nodes ◼ The probability 𝑝𝑖 that each of the 𝑚 edges connects to node 𝑖 is proportional to its current number of links 𝑘𝑖 (preferential attachment) Complex network models
- 101. 101 ◼ Barabási-Albert model (BA) Degree distribution ◼ Power-law (scale-free) with exponent = 3 Complex network models
- 102. 102 ◼ Barabási-Albert model (BA) Note ◼ Both growth and preferential attachment are needed to obtain the SF degree distribution History ◼ Yule (1925): preferential attachment to obtain SF degree distribution ◼ Simon (1955): application of modern master equation method ◼ Price (1976): application to the growth of networks ◼ Barabási & Albert (1999): rediscovery, name of preferential attachment, popularity Complex network models
- 103. 103 ◼ Barabási-Albert model (BA) Many variations ◼ Non-linear preferential attachment ◼ Dynamic edge rewiring ◼ Fitness models ◼ Hierarchic growing ◼ Deterministic growing ◼ Local growing ◼ Accelerating growth Complex network models
- 104. 104 Models according to degree distribution ◼ Erdős-Rényi model (ER) ◼ Barabási-Albert model (BA) ◼ Configuration model (CM) ◼ Interpolating model between ER and BA Complex network models
- 105. 105 ◼ Configuration model (CM) Build network given degree sequence ◼ 𝑁: number of nodes ◼ 𝑃(𝑘): degree distribution ◼ Assign a random degree 𝑘𝑖 to each node according to the given degree distribution 𝑃(𝑘) ◼ Create a vector with all the slots (“half edges”) ◼ Connect randomly pairs of slots ◼ Could be used to rewire networks ◼ Could be used to generate networks with scale-free (SF) degree distribution for any value of the exponent 𝛾 > 2 Complex network models
- 106. 106 ◼ Configuration model (CM) Algorithm Complex network models ◼ Random assignment of degrees according to P(k) ◼ Vector of slots ◼ Random permutation of the vector of slots ◼ Connection of slots
- 107. 107 ◼ Configuration model (CM) Details ◼ The number of slots must be even (2𝐿) ◼ Probability pij of connecting nodes i and j ◼ Multi-edges and self-loops may appear Complex network models
- 108. 108 ◼ Configuration model (CM) Details ◼ For large N, the number of multi-edges and self-loops is negligible if and are finite ◼ For scale-free degree distributions , the second moment diverges if the exponent is 𝛾 ∈ 2,3 Modify the algorithm to avoid the presence of multi-edges and self-loops Introduce a cut-off in P(k) scaling as ◼ Algorithms to find a random permutation Fisher–Yates shuffle (1938) Durstenfeld (1964) / Knuth (1969) Complex network models
- 109. 109 ◼ Configuration model (CM) Generalization ◼ Configuration model to build networks with desired degree- degree correlations P(k,k’) Main developers and theory ◼ Bekessy, Bekessy & Komlos (1972) ◼ Bender & Canfield (1978) ◼ Bollobás (1980) ◼ Wormald (1981) ◼ Molloy & Reed (1995) Complex network models
- 110. 110 Models according to average path length ◼ Watts-Strogatz model (WS) Complex network models
- 111. 111 ◼ Watts-Strogatz model (WS) Based on regular network and random rewirings (1998) ◼ 𝑁: number of nodes ◼ 𝑘: mean degree, even integer (𝑘 < 𝑁) ◼ 𝑝: rewiring probability ◼ Nodes initially in a regular ring lattice ◼ Each node connects to its 𝑘 nearest neighbors, 𝑘/2 on each side (clockwise and counterclockwise) ◼ For each node, rewire each clockwise original edge with probability 𝑝 to a new random destination (multi-edges and self-loops not allowed) Regular ER 0 𝑝 1 Complex network models
- 112. 112 ◼ Watts-Strogatz model (WS) Scheme Complex network models p=0.0 p=0.9 p=0.5 p=0.2 p=0.1
- 113. 113 ◼ Watts-Strogatz model (WS) Average path length and clustering Complex network models
- 114. 114 ◼ Watts-Strogatz model (WS) Conclusion ◼ Long range edges explain Small-World property Variants ◼ Adding long range edges instead of rewiring ◼ Different initial regular networks Complex network models
- 115. 115 Summary ◼ Complex networks everywhere ◼ Complex networks share common structural properties ◼ Representation of complex networks ◼ Overview of structural properties and their analysis ◼ Models of complex networks Structure of complex networks
- 116. 116 Thank you for your attention! ◼ Contact sergio.gomez@urv.cat http://deim.urv.cat/~sergio.gomez Structure of complex networks