Financial Networks: II. Fundamentals of Network Theory and FNA
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The document outlines a PhD mini-course on financial networks provided by the Center for Financial Studies at Goethe University, focusing on the fundamentals of network theory and the FNA platform. It covers command syntax, various types of graphs, network visualization, and concepts like minimum spanning trees and connectivity measures. Additionally, it includes practical examples and application areas relevant to banking and financial flows.
![Directed Weighted Graph (7)
# A directed and weighed Petersen graph.
# Create a Petersen Graph
# '-direction both' creates arcs in both direction
petersen -direction both -seed 111 -saveas WeightedDigraph -preserve false
# Set random arc property
calcap -e [?random:uniform:1,10?] -saveas value
# Visualize the network
# Setting the parameter '-arrows true' shows direction of arcs
viz -id WeightedDigraph -alabel value -awidthdefault 3 -vsizedefault 10 -
fontsize 25 -vlabel vertex_id -arrows true -saveas Digraph
# Visualize arc values as arc width
viz -id WeightedDigraph -awidth value -awidthdefault 3 -vsizedefault 10 -
fontsize 25 -vlabel vertex_id -arrows true -saveas Digraph](https://image.slidesharecdn.com/130125-financial-networks-iicfs-130126054438-phpapp01/85/Financial-Networks-II-Fundamentals-of-Network-Theory-and-FNA-14-320.jpg)
![Minimum Spanning Tree
# Create random network with 10 vertices and 30 arcs
random -nv 5 -na 10 -seed 123 -preserve false
# Calculate random arc property between 1 and 10
calcap -e [?random:uniform:1,10:111?] -saveas value
# Visualize original network
viz -vlabel vertex_id -awidthdefault 2 -saveas Random
# Identify minimun spanning tree for 'value' property
minst -p value
# Highlight Minimum Spannign Tree in 'Random' network
viz -acolor minst -alabel value -awidthdefault 2 -saveas RandomWithMST
# Drop arcs not in Minimum Spannin tree
dropa -e minst=false
# Visualize Minimum Spanning Tree
viz -vlabel vertex_id -awidthdefault 2 -alabel value -saveas MST](https://image.slidesharecdn.com/130125-financial-networks-iicfs-130126054438-phpapp01/85/Financial-Networks-II-Fundamentals-of-Network-Theory-and-FNA-21-320.jpg)
![Weighted Shortest Path (13)
# Create a Petersen Graph
# -seed 111 allows the same standard layout to be produced each time.
petersen -direction any -seed 111 -saveas Path -preserve false
#Assign random values to arcs.
calcap -e[?random:uniform:1,10:104?] -saveas value
# Calculate the path from vertex 5 to vertex 8
distance -from 00005 -to 00008 -p value -savepath true
# Visualize the network. The arc colors highlight the path
viz -acolor path(00005,00008) -alabel value -awidthdefault 2 -
vsizedefault 10 -fontsize 25 -vhover vertex_id -saveas
WeightedPath4
30](https://image.slidesharecdn.com/130125-financial-networks-iicfs-130126054438-phpapp01/85/Financial-Networks-II-Fundamentals-of-Network-Theory-and-FNA-30-320.jpg)
Financial Networks: II. Fundamentals of Network Theory and FNA
- 1. Center for Financial Studies at the Goethe University PhD Mini-course Frankfurt, 25 January 2013 Financial Networks 2. Fundamentals of Network Theory and FNA Dr. Kimmo Soramäki Founder and CEO FNA, www.fna.fi
- 2. FNA www.fna.fi
- 3. FNA Platform • Go to www.fna.fi • Register account (click login on top right) • Watch ‘Getting started with FNA’ video • More documentation available at www.fna.fi/gettingstarted 3
- 4. FNA Commands • FNA operates on commands that are submitted to FNA server for execution. Commands explore the database, alter it or create visualizations from it • Command syntax: commandname –parameter1 value1 –parameter2 value2 … e.g. loada -file sample-arcs.csv -preserve false (load arcs from sample-arcs.csv file and don’t preserve any existing networks in database) • Each command is on a single line. Character # marks a comment line • Commands can be bundled into scripts and executed in one go 4
- 5. Data Model loada -file sample-arcs.csv -preserve false sample-arcs.csv network,source,target,value 2005-1Q,Australia,Austria,499 2005-1Q,Australia,Belgium,1135 Stores the data into 2005-1Q,Australia,Canada,1884 a graph database ... on FNA Server net_id : 2005-1Q arc_id : Australia-Austria vertex_id : Austria value : 499 … vertex_id : Australia vertex_id : Belgium … vertex_id : Canada 5
- 6. Basic terms • Graph • Network = Graph with properties • Node, Vertex, (Point) • Link, (Line) • Arc = Directed Link • Edge = Undirected Link 6
- 7. Types of Networks 7
- 8. Trivial Graph (1) #G(V,E) = Trivial Graph #V = {1} #E = {empty set} # Add network ‘Trivial' to database. addn -n Trivial -preserve false # Add vertex v1 to network ‘Trivial' . addv -id Trivial -v v1 # Visualize network ‘Trivial'. viz -id Trivial -vlabel vertex_id -vsizedefault 10 -fontsize 25 -saveas TrivialViz -scale 1
- 9. Empty Graph (2) # G(V,E) = Empty Graph # V = {1, 2, 3, 4} # E = {empty set} # Add network ‘Empty' to database. # The -preserve parameter defines whethe to keep # existing networks in memory or to delete them addn -n Empty -preserve false # Add vertices v1 to v4 to network ‘Empty' . addv -id Empty -v v1 addv -id Empty -v v2 addv -id Empty -v v3 addv -id Empty -v v4 # Visualize network ‘Empty' . Add vertex names using -vlabel and change the font size of these labels using -fontsize. Set size of vertices to size 10 using -vsizedefault. viz -id Empty -vlabel vertex_id -vsizedefault 10 -fontsize 25 -saveas EmptyViz
- 10. Simple Undirected Graph (3) # G(V,E) = Simple Graph # V = {1, 2, 3, 4} # E = {(1,2),(1,3),(2,3),(2,4),(3,4)} # Add network ‘Simple' to database. addn -n Simple -preserve false # Add vertices v1 to v4 to network ‘Simple'. addv -id Simple -v v1 addv -id Simple -v v2 addv -id Simple -v v3 addv -id Simple -v v4 # Add arcs to network ‘Simple'. adda -id Simple -a v1-v2 adda -id Simple -a v1-v3 adda -id Simple -a v2-v3 adda -id Simple -a v2-v4 adda -id Simple -a v3-v4 # Visualize network ‘Simple'. viz -id Simple -vlabel vertex_id -awidthdefault 2 -vsizedefault 10 -fontsize 25 -arrows false -saveas SimpleViz
- 11. Complete Graph –K4 (4) # Add network 'netk4' to database. complete -nv 4 -directed false -preserve false # Visualize network netk4. viz -vlabel vertex_id -awidthdefault 2 -vsizedefault 10 -fontsize 25 -arrows false -saveas complete_k4 Notes If no -saveas parameter is given, created networks are autonamed Without -id paremeter in the viz - command all networks in memry are visualized
- 12. Complete Graph –K7 (5) # Add network 'netk7' to database. complete -nv 7 -directed false -saveas netk7 -preserve false # Visualize network netk7. viz -vlabel vertex_id -awidthdefault 2 -vsizedefault 10 -fontsize 25 -arrows false -saveas complete_k7
- 13. Directed Graph, Digraph (6) # Create a Petersen Graph # '-direction any' allows creation of arcs in any direction between two vertices petersen -direction any -seed 111 -saveas Digraph -preserve false # Visualize the network. Setting the parameter -arrows true gives a directed network. viz -id Digraph -awidthdefault 2 -vsizedefault 10 -fontsize 25 -vlabel vertex_id -arrows true -saveas Digraph
- 14. Directed Weighted Graph (7) # A directed and weighed Petersen graph. # Create a Petersen Graph # '-direction both' creates arcs in both direction petersen -direction both -seed 111 -saveas WeightedDigraph -preserve false # Set random arc property calcap -e [?random:uniform:1,10?] -saveas value # Visualize the network # Setting the parameter '-arrows true' shows direction of arcs viz -id WeightedDigraph -alabel value -awidthdefault 3 -vsizedefault 10 - fontsize 25 -vlabel vertex_id -arrows true -saveas Digraph # Visualize arc values as arc width viz -id WeightedDigraph -awidth value -awidthdefault 3 -vsizedefault 10 - fontsize 25 -vlabel vertex_id -arrows true -saveas Digraph
- 15. Example application areas Cross-border banking exposures Interbank payment flows 15
- 16. Bipartite Graph (8) #G(V,E) = Bipartite Graph # Vl = {1, 2, 3, 4, 5, 6, 7, 8} # Vr = {9, 10, 11, 12, 13, 14, 15, 16} # E = {randomly generated set}
- 17. Bipartite Graph # Create a bipartite graph with 8 vertices in left group (-nl) and 8 vertices in right group (-nr) and 12 randomly assigned arcs (-na) that will go in both directions. bipartite -nl 8 -nr 8 -na 12 -direction any -partition true -seed 123 -saveas bipartite -preserve false # Separate the two sets of vertices, ready for visualization. bipartitelayout -partition partition # Visualize the bipartite network. viz -vsizedefault 10 -vlabel vertex_id -awidthdefault 2 -fontsize 25 -saveas Bipartite Notes Adding the parameter -seed to the bipartite command will generate a specific graph. So for example using -seed 111 will always assign the arcs in the same way, to generate the same graph. Without the -seed parameter the the random generator is initialized from system time.
- 18. Example application areas Banks Countries Insurers Risk Drivers 18
- 19. Tree (9) # A tree is a graph with no cycles # Create a tree with 6 vertices tree -nv 10 -seed 111 -saveas Tree #Visualize the tree viz -id Tree -vlabel vertex_id -vsizedefault 5 -awidthdefault 3 Notes A forest is a disjoint union of trees
- 20. Minimum Spanning Tree (10) #G(V,E) = MST, #V = {1, 2, 3, 4, 5, 6, 7} #E = {(1,2),(1,4),(3,2),(3,4),(4,6),(4,7),(5,3),(5,4),(6,5),(6,7),(7,6)}
- 21. Minimum Spanning Tree # Create random network with 10 vertices and 30 arcs random -nv 5 -na 10 -seed 123 -preserve false # Calculate random arc property between 1 and 10 calcap -e [?random:uniform:1,10:111?] -saveas value # Visualize original network viz -vlabel vertex_id -awidthdefault 2 -saveas Random # Identify minimun spanning tree for 'value' property minst -p value # Highlight Minimum Spannign Tree in 'Random' network viz -acolor minst -alabel value -awidthdefault 2 -saveas RandomWithMST # Drop arcs not in Minimum Spannin tree dropa -e minst=false # Visualize Minimum Spanning Tree viz -vlabel vertex_id -awidthdefault 2 -alabel value -saveas MST
- 22. MST: Application Rosario Mantegna (1999). Hierarchical Structure in Financial Markets. Eur. Phys. J. B 11, 193-197. 22
- 23. Connectivity 23
- 24. Degree Degree (11) # Add network 'degree' to database petersen -direction any -seed 111 -saveas Path -preserve false # Calculate degree, in-degree and out-degree In-Degree degree -saveas degree degree -direction in -saveas in-degree degree -direction out -saveas out-degree # Visualize network degree. The vertices are labelled using their degree. viz -vlabel degree -vsizedefault 10 -vcolordefault orange - fontsize 25 -awidthdefault 3 -saveas Degree viz -vlabel in-degree -vsizedefault 10 -vcolordefault orange - Out-Degree fontsize 25 -awidthdefault 3 -saveas In-Degree viz -vlabel out-degree -vsizedefault 10 -vcolordefault orange - fontsize 25 -awidthdefault 3 -saveas Out-Degree
- 25. Degree Distribution The topology of interbank payment flows. Soramäki et al. Physica A: Statistical Mechanics and its Applications 379 (1), 317-333 25
- 26. Degree Correlations • Calculate – Neighbor degree/out-degree/in-degree (and) – Successor degree/out-degree/in-degree (asd) – Predecessor degree/out-degree/in-degree (apd) • Correlate to degree/out-degree/in-degree of each node: – zero for uncorrelated networks – positive for assortative networks and – negative fordisasortative networks • In assortative networks nodes with a given degree are more likely to have links with nodes of similar degree. Disassortative if the opposite is true. 26
- 27. Example: US Banks 27
- 28. Paths, Trails, Walks (12) # Create a Petersen Graph # -seed 111 allows the same standard layout to be produced each time. petersen -direction any -seed 111 -saveas Path -preserve false # Calculate the path from vertex 5 to vertex 8 distance -from 00005 -to 00008 -p value -savepath true # Visualize the network. The arc colors highlight the path viz -acolor path(00005,00008) -awidthdefault 2 -vsizedefault 10 - fontsize 25 -vlabel vertex_id -saveas Path Notes A Walk is any free movement along the arcs A Trail is a walk where a given arc is visited only once A Path is a walk where a given vertex is visited only once A Geadesic Path is the Shortest Path A Cycle is a path starting and ending to the same vertex
- 29. Other connectivity measures • Size: Number of arcs • Order: Number of vertices • Connectivity: order/(size*(size-1)) • Ego distance: distance to/from given vertex from/to other vertices • Eccentricity: Maximum distance from/to a vertex • Diameter: Maximum eccentricity • Clustering coefficient: Share of neighbours with links 29
- 30. Weighted Shortest Path (13) # Create a Petersen Graph # -seed 111 allows the same standard layout to be produced each time. petersen -direction any -seed 111 -saveas Path -preserve false #Assign random values to arcs. calcap -e[?random:uniform:1,10:104?] -saveas value # Calculate the path from vertex 5 to vertex 8 distance -from 00005 -to 00008 -p value -savepath true # Visualize the network. The arc colors highlight the path viz -acolor path(00005,00008) -alabel value -awidthdefault 2 - vsizedefault 10 -fontsize 25 -vhover vertex_id -saveas WeightedPath4 30
- 31. Components GWCC : Giant Weakly Connected Component GIN : Giant In-Component GSSC : Giant Strongly Connected Component Dorogovtsev S.N., J.F.F. Mendes, and A.N. Samukhin GOUT : Giant Out-Component (2001). “Giant strongly connected component of directed networks”, Phys. Rev. E 64.
- 32. Weakly Connected Graph (14) # Create random network random -nv 30 -na 40 -seed 100 -preserve false # Identify SC wc # Color vertices in SC as red setvp -p color -value black setvp -e wc=0 -p color -value red # Create visualization viz -vcolor color -vsizedefault 5 -saveas WCViZ
- 33. Strongly Connected Graph (15) # Create random network random -nv 30 -na 60 -seed 100 -preserve false # Identify SC sc # Color vertices in SC as red setvp -p color -value black setvp -e sc=0 -p color -value red # Color arcs within nodes in SC as red calcap -e source.sc=0 -saveas srcsc calcap -e target.sc=0 -saveas tgtsc calcap -e "srcsc AND tgtsc" -saveas scarc setap -p color -value black setap -p color -e scarc=true -value red # Create visualization viz -vcolor color -acolor color -vsizedefault 8 -awidthdefault 2 -saveas SCViZ
- 34. Blog, Library and Demos at www.fna.fi Dr. Kimmo Soramäki kimmo@soramaki.net Twitter: soramaki