Rich gets richer-Bitcoin Network
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Inequality in Bitcoin Network
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Rich gets richer-Bitcoin Network
- 1. 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
- 3. Bitcoin Transaction Data #Vertices : 13,086,528 #Directed Edges: 44,032,115 Transaction Data from January 2010 ~ May 2013 3/30/2015
- 4. 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
- 5. Growth of Bitcoin Network 3/30/2015 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
- 6. Yearly In Degree Distribution 3/30/2015 α= 2.18 Power Law Dist: pin(kin) ∼ kin -2.18
- 7. Yearly Out Degree Distribution 3/30/2015 Power Law Dist: pout(kout) ∼ kout -2.06 α= 2.06
- 8. Wealth Inequality : Gini Coefficient 3/30/2015 0 £ G £1 G = 0 Perfect equality 1 Complete Inequality ì í ï îï For a population uniform on the values of di
- 9. G in Bitcoin 3/30/2015 Initially Phase: G for in degree is high Users mostly stored bitc Trading Phase: Gin almost converged with Gout
- 11. Degree Distribution 3/30/2015 Degree = Indegree + Outdegree • Number of Lower degree (0~1000) nodes are very high. • Lots of isolated nodes. • Few hubs
- 12. Bitcoin Balance Statistics 3/30/2015 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
- 13. BTC balance Chart 3/30/2015 All Nodes Non-zero balance Nodes
- 14. Rich Nodes 3/30/2015 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%)
- 15. 3/30/2015 Degree Centrality: Eigenvector Centrality 3/30/2015 Which are the most important or central vertices in the network ? xi : the centrality of the node i Aij : the adjacency matrix i j Aij = 1 xi ' = Aij xj j å x' = Ax Repeat…. x(t) = At x(0) : centrality vector after t steps x(0) = civi ; vi is the eigenvector of xi i å x(t) = At civi = cili t vi = i å l1 t ci li l1 æ è ç ö ø ÷ t vi ---> c1l1 t v1 (t-->¥) i å i å Ax = l1x xi = l1 -1 Aij xj j å
- 17. Eigenvector Centrality in Bitcoin Network 3/30/2015 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 y 0.0244427 27 0.020694083 0.017641267 0.017346914 0.015898656 0.014387567 ID 3247213 3247190 3247195 3247194 3247180 3247192 Centralit y 0.0138252 13 0.013527306 0.012857685 0.01278417 0.011125401 0.010684955 …………………………. Total 13041891 centralities.
- 18. Degree Centrality: Katz Centrality 3/30/2015 xi =a Aij xj + b j 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)-1 1 (I-aA)-1 diverges when | A -a-1 I | = 0 => a-1 = l1 Therefore, a < 1 l1 Node A has eigen vector centrality 0 Unexpectedly Node B has eigen vector centrality 0
- 19. Page Rank 3/30/2015 xi =a Aij xj kj out + b j å 1 For any j , if kj out = 0 , then put kj out =1 In Matrix Term, x =aAD-1 x + b1 x = (I -aAD-1 )-1 1= D(D -aA)-1 1 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
- 20. Betweenness Centrality 3/30/2015 Measures the extent to which a vertex lies on paths between other vertices. ni st = 1 if vertex i lies between the shortest path from s to t 0 else ì í ï îï ü ý ï þï xi = ni st; st å xi = ni st s¹t å xi = ni st gst ; gst = number of shortest paths from s to t s¹t å
- 21. Closeness Centrality 3/30/2015 Measures the mean distance from a vertex to other vertices. Mean shortest path from i, li = 1 n dij j å ; dij : length of the shortest path from i to j li = 1 n -1 dij j¹i å Closeness centrality Ci Ci = 1 li Redefining, Ci = 1 n -1 1 dijj¹i å Mean shortest path of the network l = 1 n2 dij = ij å 1 n li i å
- 22. References 1. 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 3/30/2015
- 24. 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
Hinweis der Redaktion
- Figure 1. The growth of the BitCoin network. Num- ber of addresses with nonzero balance (green), addresses in participating in at least one transaction in one week intervals (red) and the exchange price of bitcoins in US dollars accord- ing to MtGox, the largest BitCoin exchange site (blue). The two black lines are exponential functions bounding the growth of the networks with characteristic times 188 and 386 days.
- Figure 2. Evolution of the indegree distribution. Since the beginning of 2011, the shape of the distribution does not change significantly. The black line shows a fitted power-law for the final network; the exponent is 2.18.
- Figure 3. Evolution of the outdegree distribution. The black line shows a fitted power-law for the the final network; the exponent is 2.06.
- Lorenz curve is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth
- After mid-2010, the degree correlation coefficient stays between −0.01 and −0.05, reaching a value of r ≈ −0.014 by 2013, In the initial phase C is high, fluctuating around 0.15 (see Fig. 5), possibly a result of transactions taking place between addresses belonging to a few enthusiasts try- ing out the BitCoin system by moving money between their addresses. In the trading phase, the clustering co- efficient reaches a stationary value around C ≈ 0.05 which is still higher than the clustering coefficient for ran- dom networks with the same degree sequence (Crand ≈ 0.0037(9))
- We provide evidence that preferential attachment is an important factor shaping these distributions. Preferen- tial attachment is often referred to as the “rich get richer” scheme, meaning that hubs grow faster than low degree nodes. In the case of BitCoin this is more than an anal- ogy: we find that the wealth of rich users increases faster than the wealth of users with low balance; furthermore, we find positive correlation between the wealth and the degree of a node.