Appendix SI - Emergence of Scale-Free Close-Knit Friendship Structure in Online Social Networks

Appendix to the manuscript.
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Table S1

The exponents of various distributions obtained by power-law fits of real online social networks and the simulated network based on the model using the maximum likelihood estimation. xmin is the lower bound of the range for fitting a power-law distribution, is the corresponding exponent and KS is the goodness-of-fit value based on the Kolmogorov-Smirnov statistic.
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Table S2

Pearson correlation coefficient. r(in; in) quantifies the tendency of nodes with a high indegree to be connected to another node with a high indegree. The other quantities carry a similar interpretation.
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Figure S1

Indegree (a) and outdegree (b) distributions of the Slashdot social network (black squares) and simulation results (red circles) based on the model. The dashed lines in both panels have a slope −2.1 as the analytic results in Eqs. (17) and (31) suggested. The simulated network is generated by the model with the parameters N =  82168, m ≈5.14 and p ≈0.67 as determined by the mean degree 〈k〉 and reciprocity of the Slashdot social network. Data points are averages over the logarithmic bins of the indegree kin and outdegree kout, respectively.
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Figure S2

Indegree (a) and outdegree (b) distributions of the Flickr social network (black squares) and simulation results (red circles) based on the model. The dashed lines in both panels have a slope −2.08 as the analytic results in Eqs. (17) and (31) suggested. The simulated network is generated by the model with the parameters N =  100000, m ≈8.07 and p ≈0.39 as determined by the mean degree 〈k〉 and reciprocity of the Flickr social network. Data points are averages over the logarithmic bins of the indegree kin and outdegree kout, respectively.
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Figure S3

Indegree (a) and outdegree (b) distributions of the YouTube social network (black squares) and simulation results (red circles) based on the model. The dashed lines in both panels have a slope −2.3 as the analytic results in Eqs. (17) and (31) suggested. The simulated network is generated by the model with the parameters N =  100000, m ≈4.34 and p ≈0.08 as determined by the mean degree 〈k〉 and reciprocity of the YouTube social network. Data points are averages over the logarithmic bins of the indegree kin and outdegree kout, respectively.
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Figure S4

Relationship between the indegree and the outdegree of nodes in the Slashdot social network and the model. Results of the Slashdot social network (black squares) and simulation results (red circles) based on the model are shown. The blue dash line represents the relation function kin = kout. Data points are averages over the logarithmic bins of the indegree kin.
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Figure S5

Relationship between the indegree and the outdegree of nodes in the Flickr social network and the model. Results of the Flickr social network (black squares) and simulation results (red circles) based on the model are shown. The blue dash line represents the relation function kin = kout. Data points are averages over the logarithmic bins of the indegree kin.
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Figure S6

Relationship between the indegree and the outdegree of nodes in the YouTube social network and the model. Results of the YouTube social network (black squares) and simulation results (red circles) based on the model are shown. The blue dash line represents the relation function kin = kout. Data points are averages over the logarithmic bins of the indegree kin.
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Figure S7

Reciprocal degree distributions of the Slashdot social network and the model. Results of the Slashdot social network (black squares) and simulation results (red circles) based on the model are shown. Analytic treatment (see Eqs. (17) and (31)) suggests a scaling behavior with an exponent −2.1, as shown by the dash line. Data points are averages over the logarithmic bins of the 13 reciprocal degree kr.
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Figure S8

Reciprocal degree distributions of the Flickr social network and the model. Results of the Flickr social network (black squares) and simulation results (red circles) based on the model are shown. Analytic treatment (see Eqs. (17) and (31)) suggests a scaling behavior with an exponent −2.08, as shown by the dash line. Data points are averages over the logarithmic bins of the reciprocal degree kr.
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Figure S9

Reciprocal degree distributions of the YouTube social network and the model. Results of the YouTube social network (black squares) and simulation results (red circles) based on the model are shown. Analytic treatment (see Eqs. (17) and (31)) suggests a scaling behavior with an exponent −2.3, as shown by the dash line. Data points are averages over the logarithmic bins of the reciprocal degree kr.
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Figure S10

Mean reciprocal degree of nodes with (a) the same indegree and (b) the same outdegree in the Slashdot social network and in the model. Results of the Slashdot social network (black squares) and simulation results (red circles) based on the model are shown in a log-log scale in the main panels. Analytic treatment suggests that 〈kr〉 is linearly dependent on kin and kout, and the blue dash lines of slope 1 show its dependence. The inset in each panel shows the results in a linear scale and the dash line has a slope of 0.82, as given by Eqs. (20) and (31). Data points are averages over the logarithmic bins of the indegree kin and outdegree kout, respectively.
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Figure S11

Mean reciprocal degree of nodes with (a) the same indegree and (b) the same outdegree in the Flickr social network and in the model. Results of the Flickr social network (black squares) and simulation results (red circles) based on the model are shown in a log-log scale in the main panels. Analytic treatment suggests that 〈kr〉 is linearly dependent on kin and kout, and the blue dash lines of slope 1 show its dependence. The inset in each panel shows the results in a linear scale and the dash line has a slope of 0.59, as given by Eqs. (20) and (31). Data points are averages over the logarithmic bins of the indegree kin and outdegree kout, respectively.
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Figure S12

Mean reciprocal degree of nodes with (a) the same indegree and (b) the same outdegree in the YouTube social network and in the model. Results of the YouTube social network (black squares) and simulation results (red circles) based on the model are shown in a log-log scale in the main panels. Analytic treatment suggests that 〈kr〉 is linearly dependent on kin and kout, and the blue dash lines of slope 1 show its dependence. The inset in each panel shows the results in a linear scale and the dash line has a slope of 0.73, as given by Eqs. (20) and (31). Data points are averages over the logarithmic bins of the indegree kin and outdegree kout, respectively.
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Figure S13

Distributions of four basic closed triples in the slashdot social network and the model. Distributions of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops in the Slashdot social network (black squares) and in the simulated network based on the model (red circles). Analytic treatment (see Eqs. (30) and (31)) suggests a scaling behavior with an exponent −1.55, as shown by the dash lines. Data points are averages over the logarithmic bins of the nFB, nFFa, nFFb and nFFc, respectively.
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Figure S14

Distributions of four basic closed triples in the Flickr social network and the model. Distributions of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops in the Flickr social network (black squares) and in the simulated network based on the model (red circles). Analytic treatment (see Eqs. (30) and (31)) suggests a scaling behavior with an exponent −1.54, as shown by the dash lines. Data points are averages over the logarithmic bins of the nFB, nFFa, nFFb and nFFc, respectively.
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Figure S15

Distributions of four basic closed triples in the YouTube social network and the model. Distributions of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops in the YouTube social network (black squares) and in the simulated network based on the model (red circles). Analytic treatment (see Eqs. (30) and (31)) suggests a scaling behavior with an exponent −1.65, as shown by the dash lines. Data points are averages over the logarithmic bins of the nFB, nFFa, nFFb and nFFc, respectively.
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Figure S16

Degree correlations in the Slashdot social network and the model. Results of degree correlations as measured by four quantities corresponding to the average nearest neighbor degree < knnin (kin) > (squares), < knnout(kin) > (circles), < knnout(kout) > (triangles), and < knnin (kout) > (inverted triangles) for (a) Slashdot social network and (b) simulated network based on the model. Data points are averages over the logarithmic bins of the indegree kin or outdegree kout.
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Figure S17

Degree correlations in the Flickr social network and the model. Results of degree correlations as measured by four quantities corresponding to the average nearest neighbor degree < knnin (kin) > (squares), < knnout(kin) > (circles), < knnout(kout) > (triangles), and < knnin (kout) > (inverted triangles) for (a) Flickr social network and (b) simulated network based on the model. Data points are averages over the logarithmic bins of the indegree kin or outdegree kout.
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Figure S18

Degree correlations in the YouTube social netowrk and the model. Results of degree correlations as measured by four quantities corresponding to the average nearest neighbor degree < knnin (kin) > (squares), < knnout(kin) > (circles), < knnout(kout) > (triangles), and < knnin (kout) > (inverted triangles) for (a) YouTube social network and (b) simulated network based on the model. Data points are averages over the logarithmic bins of the indegree kin or outdegree kout.
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Figure S19

Mean number of the four closed triples for nodes with the same indegree in the Slashdot social network and the model. Results for the mean number of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops for nodes with the same indegree are shown for the Slashdot social network (black squares) and simulated network (red circles) based on the model. Analytic treatment (see Eq. (28)) gives a scaling behavior with an exponent 2, as indicated by the dash line. Data points are averages over the logarithmic bins of the indegree kin.
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Figure S20

Mean number of the four closed triples for nodes with the same indegree in the Flickr social network and the model. Results for the mean number of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops for nodes with the same indegree are shown for the Flickr social network (black squares) and simulated network (red circles) based on the model. Analytic treatment (see Eq. (28)) gives a scaling behavior with an exponent 2, as indicated by the dash line. Data points are averages over the logarithmic bins of the indegree kin.
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Figure S21

Mean number of the four closed triples for nodes with the same indegree in the YouTube social network and the model. Results for the mean number of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops for nodes with the same indegree are shown for the YouTube social network (black squares) and simulated network (red circles) based on the model. Analytic treatment (see Eq. (28)) gives a scaling behavior with an exponent 2, as indicated by the dash line. Data points are averages over the logarithmic bins of the indegree kin.
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Figure S22

Mean number of the four closed triples for nodes with the same outdegree in theSlashdot social network and the model. Results for the mean number of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops for nodes with the same outdegree are shown for the Slashdot social network (black squares) and simulated network (red circles) based on the model. Analytic treatment (see Eq. (28)) gives a scaling behavior with an exponent 2, as indicated by the dash line. Data points are averages over the logarithmic bins of the outdegree kout.
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Figure S23

Mean number of the four closed triples for nodes with the same outdegree in the Flickr social network and the model. Results for the mean number of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops for nodes with the same outdegree are shown for the Flickr social network (black squares) and simulated network (red circles) based on the model. Analytic treatment (see Eq. (28)) gives a scaling behavior with an exponent 2, as indicated by the dash line. Data points are averages over the logarithmic bins of the outdegree kout.
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Figure S24

Mean number of the four closed triples for nodes with the same outdegree in the YouTube social network and the model. Results for the mean number of closed triples corresponding to (a) FB, (b) FFa, (c) FFb, and (d) FFc loops for nodes with the same outdegree are shown for the YouTube social network (black squares) and simulated network (red circles) based on the model. Analytic treatment (see Eq. (28)) gives a scaling behavior with an exponent 2, as indicated by the dash line. Data points are averages over the logarithmic bins of the outdegree kout.
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