Second lecture of a PhD level course on "Financial Networks" at Center for Financial Research at Goethe University, Frankfurt.

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
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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