Rich gets richer-Bitcoin Network
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Transcript of Rich gets richer-Bitcoin Network
Rich Gets Richer !
Even In Bitcoin
Network?
Announcement of Current Research State..
Zehady Abdullah Khan
Lab of Professor Hidetoshi Shimodaira
Bachelor 4th year,
Mathematical Science Course,
Department of Information and Computer Sciences,
School Of Engineering Science,
Osaka University.
3/30/2015
Bitcoin Transaction
3/30/2015
Bitcoin Transaction Data
#Vertices : 13,086,528
#Directed Edges: 44,032,115
Transaction Data from January 2010 ~ May 2013
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Unit Transaction & Network
Edges
Input AddressOutput Address
ID 1
ID 2
….
ID n_I
ID 1
ID 2
….
ID n_O 3/30/2015
4
# Possible Edge = n_I * n_O per tx
• Edges are not unique for simplicity
Growth of Bitcoin Network
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245 USD/BTC
Novermber 5, 2013
* What were the prime
Factors for
Price Increase ?
Bitcoin Exchange Started
2 Phases: 1. Initial Phase 2. Trading Phase
Yearly In Degree
Distribution
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α= 2.18
Power Law Dist: pin(kin) ∼ kin-2.18
Yearly Out Degree
Distribution
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Power Law Dist: pout(kout) ∼ kout-2.06
α= 2.06
Wealth Inequality : Gini Coefficient
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0 £G £1
G =0 Perfect equality
1 Complete Inequality
ì
íï
îï
For a population uniform on the values
of di
G in Bitcoin
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Initially Phase: G for in degree is high Users mostly stored bitcoins
Trading Phase: Gin almost converged with Gout
Pearson & Clustering
Coefficient
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Degree Distribution
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Degree = Indegree +
Outdegree
• Number of Lower degree
(0~1000)
nodes are very high.
• Lots of isolated nodes.
• Few hubs
Bitcoin Balance Statistics
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1 BTC = 383USD
(Nov 12,
2013)
Min 1st Q Median 3rd Q Max
BTC 0.00 0.00 0 0.00 111,100
USD 0 0 0 0 42,560,000
Total # of nodes Nodes with Non-
Zero Balance
Nodes with
Zero Balance
13,085,528 1,621,222 (12 %) 11464306 (88%)
Total # of BTC Total Generated BTC Left
21,000,000 11,942,900 9057100
BTC balance Chart
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All Nodes Non-zero balance Nodes
Rich Nodes
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Rich balance Nodes
> 10000 USD
Total Rich Nodes 67,512
(0.515%)
Unique Rich
Nodes
66,386
(0.507% )
Total Nodes 13,086,528
Unique Nodes 6,994,357
(53%)
Total Balance
(USD)
4,259,869,852
Rich Node
Balances
(USD)
3,960,047,047
(93%)
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Degree Centrality: Eigenvector Centrality
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Which are the most important or central vertices in the network ?
xi : the centrality of the node i
Aij : the adjacency matrixi
j
Aij = 1
xi' = Aijx j
j
å
x ' = AxRepeat….
x(t) = Atx(0) : centrality vector after t steps
x(0) = civi ; vi is the eigenvector of xii
å
x(t) = At civi = cilitvi =
i
å l1
t cilil1
æ
èç
ö
ø÷
t
vi ---> c1l1
tv1 (t-->¥) i
åi
å
Ax = l1x
xi = l1
-1 Aijx jj
å
Different Network
Measures
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Eigenvector Centrality in Bitcoin
Network
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Largest Eigen Value, l1 = 761.6418
ID 3247203 3247205 3247200 3247206 3247202 3247197
Centralit
y 1 0.083688264 0.054296171 0.050453962 0.047525502 0.032626146
ID 3247199 3247191 3247196 3247209 3247193 3247215
Centralit
y0.0244427
27 0.020694083 0.017641267 0.017346914 0.015898656 0.014387567
ID 3247213 3247190 3247195 3247194 3247180 3247192
Centralit
y0.0138252
13 0.013527306 0.012857685 0.01278417 0.011125401 0.010684955
…………………………. Total 13041891 centralities.
Degree Centrality: Katz Centrality
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xi = a Aijx j + bj
å
; a, b are positive constants
Here a keeps the balance.
x = aAx + b1 ; 1=(1,1,1,...)
a -> 0 , x =b1
x = (I-aA)-11
(I-aA)-1 diverges when
| A-a -1I | = 0 => a -1 = l1
Therefore, a <1
l1
Node A has eigen vector centrality 0
Unexpectedly Node B has eigen vector
centrality 0
Page Rank
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xi =a Aijx j
k jout
+ bj
å 1
For any j , if k jout = 0 , then put k j
out =1
In Matrix Term,
x = aAD-1x + b1
x = (I -aAD-1)-11= D(D-aA)-11
a < l, where l is the largest eigenvalue of AD-1
For undirected graph,
l = 1 with eigenvector v = (k1,k2,k3,....)
If a vertex with high Katz centrality points to a large number of other vertices, all
those vertices will gain high centrality !!!!
i
All other nodes gains high
centrality if the node i
has high Katz centrality
Betweenness Centrality
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Measures the extent to which a vertex
lies on paths between other vertices.
nist =1 if vertex i lies between the shortest path from s to t
0 else
ìíï
îï
üýï
þï
xi = nist; st
å
xi = nist s¹t
å
xi =nist
gst ; gst = number of shortest paths from s to t
s¹t
å
Closeness Centrality
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Measures the mean distance from a vertex to other vertices.
Mean shortest path from i,
li =1
ndij
j
å ; dij : length of the shortest path from i to j
li =1
n-1dij
j¹i
å
Closeness centrality Ci
Ci =1
li
Redefining,
Ci =1
n-1
1
dijj¹i
å
Mean shortest path of the network
l =1
n2dij =
ij
å1
nli
i
å
References1. Do the rich get richer? An empirical analysis of the BitCoin transaction
network. MIT tech
Daniel Kondor,∗ Marton Posfai, Istvan Csabai, and Gabor Vattay ,Department of Physics of Complex Systems,Eotvos Lorand University, Hungary
2. Networks An Introduction
M.E.J Newman
3. Albert, R. and Barabasi, A.-L. (2002). Statistical mechanics of complex networks. Reviews of modern physics 74 (1), 47
4. Quantitative Analysis of the Full Bitcoin Transaction Graph. http://eprint.iacr.org/2012/584
Ron, D. and Shamir, A. (2012).
5. http://bitcoin.org/about.html
6. http://www.vo.elte.hu/bitcoin
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The End
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What’s Next ?
Detail Dynamics of transaction
Non-parametric estimation of preferential
attachment function.
Network Visualization of important Bitcoin entity
Extracting Interesting Bitcoin Phenomenon
Bitcoin price prediction.
3/30/2015