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.

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Bitcoin Transaction

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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.

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