Fitting a Normal Distribution - Friendship Evolution and the Three-Rule Model

Chapter 3 Friendship Evolution and the Three-Rule Model

3.4. Fitting a Normal Distribution

For sensitivity analyses of skewness and critical parameters affecting distribution, a feasible probability-distribution function (pdf) must be applied. In most situations a

normal distribution is considered the best choice, but it does not fit our purposes in this study. Since critical parameters such as initial friendship, old friends remembering, resources, and breakup thresholds have ranges of 0 to 1, we chose a beta distribution—a two-parameter family of continuous probability distributions defined according to the interval [0, 1] with a probability density function of

0.2 0.4 0.6 0.8 1.0

Figure 3. Comparison of beta and normal distributions.

Once a simulation reaches a statistically stationary level, then clustering coefficient, average path length, average degree, average of degree squared, and degree distribution statistics can be collected. Degree distributions in our simulations involved some random rippling, especially for smaller populations. However, since large populations consume dramatically greater amounts of simulation time, we applied Bruce’s (2001) ensemble average as follows:

∑

where M is the number of curves to be averaged and p(k) a curve that represents, for example, a degree distribution, a time series, and so forth.

Chapter 4 Experiment

A simulation of our model begins with parameter initialization and ends once the acquaintance network reaches a statistically stationary state. Initialized parameters included the number of persons N, leaving and arriving probability p, updated friendship proportion b, old friend remembering q, breakup threshold

, distribution of friend-making resources r, and distribution of initial friendship f₀. Statistically stationary states were determined by observing average degree <k>, average square of degree <k²>, clustering coefficient C, and average path length L. Each of these four statistics eventually converged to values with slight ripples.

A statistically stationary state of parameter initialization at N = 1,000, p = 0, b = 0.001, q = 0.9,

= 0.1, r with a fixed value of 0.5, and a beta14 f0 ( = 0.9) is shown in Figure 4. Solid lines indicate the acquaintance network and the dashed lines in Figures 4c and d (which are calculated using Equation 6 and Equation 7) indicate the ER random model at the same average degree as the acquaintance model.

0 10 20 30 40 50 60 70 80

Figure 4. Example of a statistically stationary state using the proposed model.

4.1. Effects of Leaving and Arriving

For comparison, we reproduced Davidsen et al.’s (2002) simulations using their original parameters of N = 7,000 and p at 0.04, 0.01, and 0.0025. We then changed N to 1,000 and tested a broader p range. As noted in an earlier section, the leaving and arriving probability p is the only parameter in rule 2. In addition to using various degree distribution diagrams, we gathered <k>, C, and L varying in p and analyzed their correlations to determine the effects of p on the acquaintance network.

The degree distribution P(k) from the two-rule model is shown in Figure 5. All <k>,

C, and L values with parameter initializations for various probability p values are shown in Figure 6. Correlations among <k>, C, and L are shown in Figure 7. The solid lines in Figure 6 reflect the application of Davidsen et al.’s two-rule model; the dashed lines (which are calculated using Equation 6 and Equation 7) reflect the application of the ER model at the same average degree. Contrasts between the two lines in Figures 6b and 6c indicate that the acquaintance network has the small world characteristic. Figure 6a shows that the number of friends increases as the lifespan of an individual lengthens.

According to Figure 7d, the clustering coefficient closely follows average degree not but average path length.

A larger p indicates a higher death rate and a lower p a longer life span. Thus, parameter p acts as an aging factor. Relative to other species, humans require more time to make friends; Davidsen et al. therefore only focused on the p << 0.1 regime. To satisfy the needs of integrity theory, we also explored the p >> 0.1 regime and found that mean degree <k> decreased for p values between 0 and 0.5. The decrease slowed once p > 0.1 (Fig. 6a).

10⁰ 10¹ 10² 10³

k

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

P (k)

p=0.5 p=0.1 p=0.04 p=0.01 p=0.0025

10⁰ 10¹ 10² 10³

k

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

P (k)

p=0.5 p=0.1 p=0.04 p=0.01 p=0.0025

Figure 5. Two-rule model degree distribution P(k).

Table 1. <k>, C and L vary in leaving and arriving probability p.

p .5 .1 .05 .04 .01 .0025

<k> 1.55 5.246 10.02 12.64 39.71 115.34

< k²> 4.88 60.13 314.18 467.23 4708.25 29652.69

C 0.213 0.413 0.453 0.465 0.577 0.697

L 17.076 4.226 3.262 3.053 2.416 2.111

0.0 0.1 0.2 0.3 0.4 0.5

Figure 6. <k>, C and L vary in leaving and arriving probability p.

0 20 40 60 80 100 120

Figure 7. Correlations among <k>, C and L vary in leaving and arriving probability p.

Note that in our model, a leaving and arriving probability of 0 means that rule 2 is inactive, and a friendship update proportion of 0 means that rule 3 is inactive. Once rule 3 becomes inactive, our three-rule model becomes the equivalent of Davidsen et al.’s two-rule model. In all of the experiments described in the following sections, N was initialized at 1,000 and b at 0.001.

4.2. Effects of Breakup Threshold

To determine the effects of the breakup threshold on the acquaintance network, experiments were performed with parameters initialized at different levels of the

friendship-breakup threshold . Other initialized parameters were q = 0.6 and the constants r = 0.5 and f0 = 0.5. The solid lines in Figure 8 represent <k>, C, and L statistics without rule 2 (p = 0) and the dashed lines represent the same statistics with rule 2 included (p = 0.0025). The data indicate that rule 2—which acts as an aging factor on acquaintances in the network—reduced both average degree <k> and clustering coefficient C and increased average path length L.

According to the data presented in Figure 8, the breakup threshold ^θ lowers the average degree <k> and raises both the clustering coefficient C and average path length L.

The Figure 9 data show that the C–<k> and L–<k> corrections are negative and the C–L correction is positive. The threshold reflects the ease with which a friendship is broken.

As expected, a higher ^θ results in a smaller number of “average friends” and greater separation between individuals.

Table 2. <k>, C and L vary in breakup threshold ^θ with different leaving and arriving probability p.

0.2 0.1 0.05 0.025 0.0125

p 0

<k> 5.99 9.94 14.39 22.05 38.21

< k²> 58.84 153.62 256.06 533.08 1528.89

C 0.3778 0.3146 0.1899 0.1284 0.1085

L 4.0986 3.2369 2.8368 2.5431 2.1414

p 0.0025

<k> 6.00 9.61 14.30 21.89 34.84

< k²> 60.48 146.94 257.08 531.31 1365.39

C 0.3876 0.3112 0.1532 0.1108 0.1484

L 4.0543 3.3064 2.8975 2.6230 2.3075

0.02 0.04 0.06 0.08 0.10

0.10

0.02 0.04 0.06 0.08 0.10

0.10

0.02 0.04 0.06 0.08 0.10

Figure 8. <k>, C and L vary in breakup threshold ^θ with different leaving and arriving probability p

5 10 15 20 25 30 35 40

Figure 9. Correlations among <k>, C and L vary in breakup threshold ^θ.

4.3. Effects of Resources

To determine the effects of resources and memory factors on acquaintance networks, we ran a series of experiments using parameters initialized with different friend-making resource r and friend-remembering q values. Initialized parameters also included p = 0, ^θ

= 0.1, and a fixed f0 value of 1. According to our results, a larger r raised the average degree <k> but lowered the clustering coefficient C and average path length L (Fig. 10).

While it is not obvious that statistical characteristics are influenced by different resource

distributions. The Figure 10 data also show that an increase in q raised <k> and lowered both C and L. Furthermore, C–<k> and L–<k> corrections were identified as negative while the C–L correction was positive (Fig. 11).

Table 3. <k>, C and L vary in friend-remembering q value with different distributions of friend-making resource r.

q 0. .2 .4 .6 .8 .9

r beta14(=0.1)

<k> 4.00 4.05 6.00 8.17 18.00 39.95

< k²> 24.62 25.60 68.48 135.10 616.24 2338.91

C 0.3588 0.3816 0.3582 0.3590 0.2969 0.1780

L 0.5254 5.4701 3.9116 3.4143 2.7518 2.2644

r beta14(=0.5)

<k> 4.96 6.03 7.99 11.43 21.20 41.25

< k²> 30.59 49.21 105.52 237.55 753.64 2438.32

C 0.4554 0.3840 0.3642 0.3245 0.2382 0.1802

L 6.0173 4.4481 3.4747 3.0483 2.5915 2.2003

r beta14(=0.9)

<k> 8.18 9.73 11.60 14.00 23.97 43.99

< k²> 71.30 109.03 161.57 252.06 821.56 2625.60

C 0.1905 0.2246 0.1737 0.1778 0.1705 0.1461

L 3.8626 3.3340 3.1260 2.8912 2.4840 2.1564

r fixed_value(=0.5)

<k> 4.68 6.02 8.01 11.37 19.97 40.30

< k²> 26.03 46.23 95.12 226.93 659.40 2356.56

C 0.486 0.386 0.348 0.319 0.275 0.189

L 6.954 4.611 3.575 3.067 2.624 2.201

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 10. <k>, C and L vary in friend-remembering q value with different distributions of friend-making resource r.

0 5 10 15 20 25 30 35 40 45

Figure 11. Correlations among <k>, C and L from the different distributions of friend-making resource r.

4.4. Effects of Initial Friendship

Experiments were run using parameters initialized at different initial-friendship f0 and friend-remembering q values for the purpose of determining the effects of those factors on acquaintance networks. Other initialized parameters were p = 0, ^θ = 0.1, and a fixed r value of 0.5. Our results show that a larger f0 raised the average degree <k> but lowered both the clustering coefficient C and average path length L (Fig. 12). That different distributions of initial friendship influenced the statistical characteristics was not obvious, but different averages of initial friendship clearly did. In other words, <k>, C, and L were affected by different initial friendship averages but not by different initial friendship

distributions. The Figure 12 data also show that the friend remembering q factor raised

<k> and lowered both C and L. Both C–<k> and L–<k> corrections were negative and the C–L correction positive (Fig. 13).

Table 4. <k>, C and L vary in friend-remembering q value with different distributions of initial friendship f0

q 0. .2 .4 .6 .8 .9

f0 beta14(=0.1)

<k> 4.01 4.14 4.58 5.05 6.00 8.00

< k²> 19.00 19.59 24.33 30.8 44.01 84.36

C 0.5007 0.4788 0.4539 0.4512 0.3547 0.2886

L 9.2204 9.7683 7.7325 6.6268 4.7783 3.7074

f0 beta14(=0.5)

<k> 4.74 6.00 7.03 9.52 15.96 29.98

< k²> 25.41 47.06 69.95 141.19 442.85 1342.34

C 0.4892 0.4023 0.3652 0.3232 0.2939 0.2065

L 8.1108 4.5738 3.8898 3.2932 2.7749 2.3597

f₀ beta14(=0.9)

<k> 4.66 5.99 7.99 10.59 20.04 40.04

< k²> 25.24 45.92 92.82 179.43 625.14 2330.23

C 0.5064 0.4068 0.3370 0.3031 0.2313 0.1691

L 8.2640 4.7841 3.6030 3.1511 2.6172 2.2277

f0 fixed_value(=0.5)

<k> 4.66 6.00 7.37 9.94 16.01 30.46

< k²> 24.85 46.29 76.41 153.62 395.63 1352.15

C 0.4950 0.4299 0.3670 0.3146 0.2312 0.1862

L 7.9614 4.8324 3.8830 3.2369 2.7998 2.3541

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 12. <k>, C and L vary in friend-remembering q value with different distributions of initial friendship f₀.

0 5 10 15 20 25 30 35 40 45

Figure 13. Correlations among <k>, C and L from the different distributions of initial friendship f₀.

We analyzed the effects of different parameters on our proposed model by relationally cross-classifying all experiments; our results are shown in Tables 5 and 6.

The plus/minus signs in Table 5 denote positive/negative relations between parameters and statistics. In Table 6 the plus or minus signs denote the strength and direction of correlations. As Table 5 indicates, in addition to the effects of rule 1, q, r, and f0 had positive correlations with average degree <k> and p and ^θ had negative correlations with

<k>. Furthermore, each average degree had a negative relationship with its corresponding average path length. All of the rule 3 parameters affected the clustering coefficient C and average length L in a positive manner, while rules 1 and 2 affected C and L negatively. Note that friendships are initialized in rule 1 and updated in rule 3.

Table 5. Effective directions of the parameters on <k>, C, L.

Rule 2 Rule 3

Statistics Rule 1

p q ^θ r f0

<k> +^* - + - + +

C +^* - - + - -

L -^* + - + - -

Table 6. Summary of the correlations between <k>, C, L from above experiments

Experiments Variational

Parameters C-<k> L-<k> C-L

4.1 p +++ -- ---

4.2 ^θ, p -- --- +++

4.3 q, r --- -- +++

4.4 q, f0 --- -- +++

4.5. Sampling

Surveys, questionnaires, and sampling techniques stand at the center of traditional social science research and are considered cheaper and more practical than collecting large amounts of census data. However, the effectiveness of these methods for analyzing social networks has not been examined. We therefore ran an arbitrary simulation of our model after reaching a statistically stationary state and collected a sample of nodes. Initialized parameters were N = 1,000, p = 0, b = 0.001, q = 0.4, and ^θ = 0.1; constants were r = 0.5 and f0 = 0.5.

Figure 14 presents the degree distribution P(k) after sampling at 100, 300, 500 and 700 nodes. Figures 14a and 14b are log plots with log scaling on the x and y axes; these

were used to determine if distributions were scale-free. Figures 14c and 14d are semi-log plots with log scaling on the x axis only; these were used to determine if distributions were exponential. Degrees in Figures 14b and 14d are post-normalization, as required for different numbers of sampled nodes. Each curve in Figure 14 represents an ensemble average of 100 sampling repetitions. The solid lines in Figure 14 reflect a lower sampling ratio of 0.1—considered common for traditional surveys and sampling techniques. The dotted lines reflect a higher sampling ratio (0.7) considered common for a census. Turns in the direction of the y-axis were observed for high sampling but not for low. The degree distribution clearly lost its original shape after sampling.

Chapter 5 Conclusion

Most small world models of social networks are analyzed by mixing regular graphs with random networks. This practice is based on Watts and Strogatz’s (1988) model, which mixes one regular and one random graph to facilitate theoretical analysis. While the WS model has had a strong impact, there is growing awareness that social network research should not be restricted to issues associated with separation and clustering (Newman &

Park, 2003).

Exploring how people make new friends is a meaningful task. In most cases, people make new social connections via introductions by friends in common, but there are many cases in which strangers become friends through chance meetings with no introductions.

With few exceptions, most of us can only give limited attention or spend limited resources on friend-making, therefore friends who were once considered close can become distant over time. To gain a better understanding of acquaintance networks, we propose a three-rule model of network evolution. In rule 1, acquaintances are made via introductions and chance meetings; an aging factor is added in rule 2; in rule 3, friendships are altered according to such factors as limited resources, friend remembering, breakup thresholds, and initial friendships.

In our model, small world statistics (especially mean degree for each node) were solely dependent on the average for each parameter. For example, we used a fixed

friend-making resource value to compare resources with beta14 distributions of different averages, and found that the <k> statistic says more about average than resource distribution. A similar phenomenon was also found for the initial friendship factor f₀.

Experimental simulations are a necessary aspect of social network research, not only because of the expenses and other difficulties involved with fieldwork, but also because widely used sampling approaches cannot capture real social network distributions, since distributions for higher sampling rates differ from those for lower sampling rates.

Taking a bottom-up, human-interaction-based simulation approach to modeling is a reflection of the evolution mechanism of real social networks. Building on insights from previous studies, we applied local and interactive rules to acquaintance network evolution. This approach produced new findings that can be used to explore human activity in specific social networks—for example, rumor propagation and disease outbreaks.

Appendix A

Terms and Abbreviations

Table 7. Terms and abbreviations for parameters.

Abbreviation Description

N Number of persons (nodes) in acquaintance network. （結點數、人數）

M Number of friendships (edges) in acquaintance network. （友誼數、邊數）

f0, f(t=0) Initial friendship distribution. （初始友誼）

p Leaving and arriving probability in rule 2.（出入機率）

b Proportion of updated friendships in rule 3.（友誼更新比）

f Friendships. (友誼深淺)

q Old friend remembering.（念舊度）

r Friend-making resource distribution. （交友資源）

θ, th Breakup threshold. （斷交門檻）

, mu Mean.（平均值）

Table 8. Terms and abbreviations for statistics.

Abbreviation Description

<k> Average degree. （度均值）

<k²> Average square of degree. （度方均值）

C Average clustering coefficient. （群聚係數）

L Average shortest path length. （平均最短路徑長度）

Table 9. Terms and abbreviations for initial distributions.

Abbreviation Description

fixed_value( ) A distribution that sets its random variable as a fixed value .

beta14( ) A beta distribution instance in [0, 1] that forces ^α + ^β = 14 with = ^α / (^α + ^β).

Appendix B

Application to Examining Scales Effects

A model acquaintance network can be used to examine the effects of network scales. An arbitrary simulation of our model after reaching a statistically stationary state was selected to sample its maximum connected components for the purpose of selecting a connected subgraph with a specific number of nodes. The selected network contained five connected components, with the maximum connected components holding 986 of 1,000 nodes. Selected acquaintance network parameters were initialized at N = 1,000, p = 0, b = 0.001, q = 0.4,

= 0.1, r = 0.5 (fixed) and f0 = 0.5 (fixed).

Figure 15 shows the degree distribution P(k) after sampling for selecting a connected subgraph with the number of nodes set at 100, 300, 500 and 700. Subfigures a and b are log plots with log scaling on the x and y axes (for determining if the distribution is scale-free). Subfigures c and d are semi-log plots with log scaling on the x axis only (for determining if the distribution is exponential). Degrees for subfigures b and d are post-normalization. Each curve in the figure is from an ensemble average of 100 sampling repetitions.

Figure 15. Model acquaintance network scales.

Appendix C

Application to Epidemiology

Some infections (e.g., common colds) do not confer long-lasting immunity. Accordingly, such infections do not have a “recovered” state and individuals become repeatedly susceptible. We applied our model acquaintance network to a simulation of the SIS model (SIS stands for Susceptible, Infected, and Susceptible). An acquaintance network and its analogous NW model were selected to make comparisons of time series and phase transitions with the SIS model.

Selected acquaintance network parameters were initialized at N = 1,000, p = 0, b = 0.001, q = 0.4,

= 0.1, r = 0.5 (fixed) and f0 = 0.5 (fixed). Table 10 compares the statistics between an instance of our acquaintance network and its analogous NW model. Table 11 lists the SIS parameters using for Figure 16 and Figure 17. Figure 16 shows the time series during SIS model simulation process. Figure 17 shows the phase transitions of SIS model.

Table 10. Network statistics.

Our Acquaintance

Network

Analogous NW Model

Mean of degree <k> 7.844 8.092 Clustering coefficient C 0.346 0.346 Average shortest path length L 3.712 4.109

Table 11. SIS model parameters.

Figure 16 Figure 17 Infection rate 0.13 from 0 to 0.4 in

rules of 0.02

Recovery rate 0.9 1

Initial ratio of infected people 0.005 0.5

Repetitions 20 20

Figure 16. Time series during SIS model simulation process.

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Infection Transmission Rate, λ

0.0 0.1 0.2 0.3 0.4 0.5

F in a l In fe ct ed R a ti o , ρ

3-rule Acq.

NW model

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Infection Transmission Rate, λ

0.0 0.1 0.2 0.3 0.4 0.5

F in a l In fe ct ed R a ti o , ρ

3-rule Acq.

NW model

Figure 17. SIS model phase transition.

Appendix D

Distribution of Co-directors

The three main properties of social networks are (a) the small-world phenomenon, (b) the high-clustering characteristic, and (c) skewed degree distribution. In this chapter, we focus on the third property. Networks of board and director interlocks reveal a remarkable degree distribution that is different by far from either scale-free or normal random networks [24]. For example, the nearly 8000 directors on the board of Fortune 1000 companies in 1999 are connected, and the corresponding degree distribution has a strongly peak and a fast approximately exponential decay in the tail, much faster than a power-law distribution but slower than a Poisson or normal distribution [see also 19 and 21].

Figure 18 shows the degree-distribution comparison between one of our acquaintance networks after at a statistically stationary state (solid curve) and co-directors for Davis’

boards-of-directors data (dashed curve). Selected acquaintance network parameters were initialized at N = 1,000, p = 0, b = 0.001, q = 0.4, ^θ = 0.1, r = 0.5 (fixed) and f₀ = 0.5 (fixed). Davis’ data is about the nearly 8000 directors on the board of Fortune 1000 companies in 1999 [23]. Both curves exhibit distinct peaks and then a long tail that doesn’t appear to decay smoothly.

Figure 18. Degree distribution comparison with Davis’ boards of directors data.

Appendix E Source Code

The simulation was coded in Python using the NetworkX package. Statistical data collection was performed using Python with the matplotlib package. Information about Python and relative package installation is available at http://yukuan.blogspot.com/2006/08/graph-based-modeling-on-python.html. Table 12 briefs the files in our acquaintance network project.

Table 12. Summary of source files.

Directory File Name Description

./src/ acq2006.py Our three-rule model.

./src/ acq2002.py Davidsen et al.’s two-rule model.

./src/acq2006log/ *.log Log files for experimental simulations of our model.

./src/acq2002log/ *.log Log files for experimental simulations of Davidsen et al.’s model.

./src/ *.adjlist Files for storing acquaintance networks in stationary states using adjacency-list.

./src/acq2006log/ stats_fig.v1.8.py For generating all other figures.

./src/acq2006log/ base.py For parsing log files.

./src/acq2006log/ acq2002Figs.py For generating Figures 5, 6 and 7.

./src/acq2006log/ plotKCL.py For generating Figures 8, 10 and 12.

./src/acq2006log/ plotCorr.py For generating Figures 9, 11 and 13.

./src/ pdf_fig.py For generating Figures 2 and 3.

./src/ sis.py For application to SIS model (Figs. 16 & 17).

./src/ sampleNode.py For sampling an acquaintance network to select a sub-net with k nodes (Fig. 14).

./src/ sampleMaxCmp.py

For sampling max connected component of an acquaintance network to select a connected subgraph with k nodes (Fig. 15).

./src/ components.py To gather statistics for connected components.

./src/ assortCoef.py To gather statistics of Assortativity Coefficients ./src/ direct99.csv Gerald (Jerry) Davis’ boards of directors data ./src/ direct99.py To process Davis’ boards of directors data

./src/ acqCk.py To plot C-k diagram for our acq. net.

./src/ Davis99AcqCmp.py For generating Figure 18

Appendix F

"""Generate random variates with beta distribution

and satisfies alpha+beta=14.

11 12

While mu=.5 The distribution is similar to normal

distribution with mu=.5, sigma=.136

14 15

ref. http://en.wikipedia.org/wiki/Beta_distribution

ref. http://docs.python.org/lib/module-random.html

"""Initialize the acquaintance network.

"""Friend making of two persons.

38 39

One randomly chosen person picks any two his friends and introduces

them to each another. If they have not met before, a new link

between them is formed. In case the person chosen has less than two

acquaintances, he introduces himself to one other random person.

"""

u, v = rnd.sample(_G.nodes(), 2)

With probability p, one randomly chosen person is removed from the

network, including all links connected to this node, and replaced by

def _update_friendship(u, v):

"""Friend remembering between two persons.

67 68

The two persons update their friendship via the following:

- f_new = q*f_old + (1-q)*J(D(r_u/k_u), D(r_v/k_v))';

- The friendship breaks when f_new is less than a threshold.

"""Friend remembering for any b*M friendships.

90 91

M means the total friendships/edges of the whole network.

"""Loop the three steps of Iters times and gather statistics.

101 102

"""

(For details please see acq2006.py)

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在文檔中資源和記憶對舊識網路的影響 (頁 27-0)