sex differences in intimate relationships › pdf › 1201.5722v1.pdf · mating systems, the...

Sex differences in intimate relationships Vasyl Palchykov1, Kimmo Kaski1, János Kertész1,2, Albert-László Barabási2,3 and

Robin I.M. Dunbar4

1Department of Biomedical Engineering and Computational Science (BECS), Aalto University School of Science, P.O. Box 12200, FI-00076, Finland;

2Institute of Physics BME, Budapest, Budafoki ut 8., H-1111; 3Center for Complex Networks Research, Northeastern University, Boston, MA 02115;

4Institute of Cognitive & Evolutionary Anthropology, University of Oxford, 64 Banbury Road, Oxford

OX2 6PN, UK

Social networks have turned out to be of fundamental importance both for our understanding human sociality and for the design of digital communication technology. However, social networks are themselves based on dyadic relationships and we have little understanding of the dynamics of close relationships and how these change over time. Evolutionary theory suggests that, even in monogamous mating systems, the pattern of investment in close relationships should vary across the lifespan when post-weaning investment plays an important role in maximising fitness. Mobile phone data sets provide us with a unique window into the structure of relationships and the way these change across the lifespan. We here use data from a large national mobile phone dataset to demonstrate striking sex differences in the pattern in the gender-bias of preferred relationships that reflect the way the reproductive investment strategies of the two sexes change across the lifespan: these differences mainly reflect women’s shifting patterns of investment in reproduction and parental care. These results suggest that human social strategies may have more complex dynamics than we have tended to assume and a life-history perspective may be crucial for understanding them.

Keywords: human behaviour; sex difference; social networks; social communication technology.

1. INTRODUCTION 1

Social relationships, and in particular pairbonds, 2 are the outcome of individuals’ decisions about whom 3 to invest their available social capital in. Such decisions 4 typically reflect a choice between the payoffs offered by 5 alternative candidates. However, in monogamous 6 species, and especially those that live in multi- 7 generational families, investment strategies may vary 8 across the lifespan as a function of the individual’s 9 changing reproductive circumstances – notably the 10 impact of constraints such as the risk of death or the 11 cessation of active reproduction [1,2]. In species like 12 humans, where menopause truncates female 13 reproductive activity and investment in offspring 14 typically continues into adulthood, evolutionary theory 15 would predict that investment in relationships should 16 vary across the lifetime as a function of the trade off 17 between the relative opportunities for personal 18 reproduction versus (grand-)parental investment. In this 19 respect, evolutionary theory would also predict 20 significant contrasts in the social strategies of the two 21 sexes as a function of the differences in their 22 reproductive strategies. However, studying human 23 social relationships in any detail on a large scale has 24 proved unusually difficult. The bottom-up approach 25 adopted by social psychologists and sociologists has 26 commonly been limited by sample size, while the more 27 recent top-down social network analysis approach 28 inevitably suffers from a lack of detail about the 29

individuals involved [3,4]. More importantly, most 30 large-scale network studies have tended to treat 31 relationships as static, and ignore the fact that social 32 relationships are dynamic and change over time, at the 33 very least on the scale of a lifetime. 34

In humans, homophily (a tendency for individuals 35 who share traits to preferentially form relationships) has 36 emerged as an important organizing principle of social 37 behaviour [5,6]. In most such cases, studies of 38 homophily have focussed on psychological or social 39 traits such as personality, interests, hobbies, and 40 religious or political views. However, there is evidence 41 that homophily may also arise through a tendency for 42 close friendships to be gender-biased [7,8] and we 43 exploit this to explore the changing patterns of 44 relationship investment across the lifespan. We use a 45 cross-sectional analysis of a very large mobile phone 46 database to investigate gender preferences in close 47 friendships: i) to test the hypothesis that preferences in 48 the choice of the “best friend” are gender-biased 49 (homophilic with respect to gender); and ii) to 50 investigate how these preferences change over the 51 lifespan. We focus our attention on the three most 52 preferred friends, as indexed by the frequency of 53 contact. Several studies have demonstrated that 54 frequency of contact is a reliable index of emotional 55 closeness in relationships [9,10]. Recent research also 56 reveals that personal social networks are hierarchically 57 structured [11,12], having a layer-like structure with 58

distinct differences in emotional closeness and 59 frequency of contact with alters in the different layers, 60 with an inner core of ~ 5 alters who between them 61 account for about half our total social time [9,13]. 62

63 2. METHODS AND RESULTS 64

For our study we used the large-scale hashed 65 mobile phone dataset from a single mobile service 66 provider in a specific European country [14–16] to 67 investigate the age and gender correlations between sets 68 of “best friends”. In order to filter out spurious effects 69 like accidental or wrong number events as well as 70 professional (e.g. call centre) calls, we considered calls 71 and text messages only between individuals that had at 72 least one reciprocated contact (i.e. a contact in each 73 direction). Among the 6.8 million subscribers of this 74 provider who meet this criterion, we consider only those 75 whose gender and age are both known, and for whom 76 only a single subscription is registered. These account 77

million subscribers, of whom about 1.8 78 million are males and about 1.4 million females. The 79 dataset covers a seven-month period and includes 1.95 80 billion calls and 489 million text messages. Finally, we 81 performed some additional data filtering to remove 82 obviously erroneous records in the dataset (for details, 83 see the electronic supplementary material (ESM)). 84

We define the “best friend” of a given subscriber 85 i as the alter that i is most frequently in contact with, 86 counting both the number of calls and text messages; 87 the “second best friend” is then the alter that i is in 88 contact with next most frequently, and the “third best 89 friend” is the next most frequently contacted individual, 90 etc. Restricting ourselves to pairs of subscribers for 91 both of whom we have age and gender information 92 gives us 1.19 million ego/best-friend pairs, 0.80 million 93 ego/second-best-friend pairs and 0.66 million ego/third- 94 best-friend pairs. These numbers of ego/friend pairs are 95 smaller than what would come out if the total number 96 of subscribers in the sample could be used. The 97 reduction is related to the described restrictions with 98 the dataset and may introduce uncorrelated 99 randomness, which we, however, do not expect to have 100 any significant influence on the main conclusions of 101 this study. 102

For our analysis, we identify the gender of each 103 subscriber by the variable , such that 104

for males and for females. We define 105 the average gender as , where the 106 summation is taken over all subscribers. Since 107

for the whole dataset, there is an imbalance 108 in favour of male alters. The average gender of the 109 ego’s “best friend” is defined as , where 110 stands for the gender of the “best friend” of subscriber i 111 and in the summation i runs over pairs of subscribers 112 with known gender and age information. With the 113 overall , the “best friends” are almost 114

perfectly balanced. Compared that to 115 indicates that there is a strong bias in the selection due 116 to some kind of gender correlation. (The balance of the 117 egos and the best friends is depicted as a function of the 118 age of the egos in the ESM A. Dataset). Considering 119 male and female subscribers separately, we reveal that 120 the “best friends” are usually characterized by opposite 121 genders with the average gender of the “best friend” 122 having an overall value for males and 123

for females. 124 In order to determine what this gender correlation 125

is, we examine the average gender of the “best friend” 126 as a function of ego’s age, for male and female egos 127 separately (Fig. 1a). It is apparent that until the age of 128 about 50 years, both male and female egos prefer their 129 “best friend” to be of the opposite gender, although this 130 effect is strongest for 32 year old males and 27 year old 131 females, yielding peak values of and 132

for males and females, respectively. 133

Notice that not only does the preference for an 134 opposite-sex “best friend” kick in significantly earlier 135 for females than for males (~18 years vs mid-20s, 136 respectively), but females maintain a higher plateau 137 value for much longer. Males exhibit a distinct and quite 138 short-lived peak (of about 7 years as indicated by a 139 20 % decline from the peak value at 32 years of age); in 140

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Figure 1. Gender correlations between best friends. A: Average gender of the “best friend” of an ego of specified age and gender (n male, � female). B: Average gender of the “second best friend” of an ego of specified age and gender (n male, � female). Note that these results are overall independent of the definition of the “best friend” (see ESM).

contrast, women have a long, relatively higher male- 141 biased plateau (of about 14 years, as defined by the 142 onset of a 20 % decline from the peak at 27 years of 143 age), after which the male “best friend” seems to be 144 moved to the second place (Fig. 1b) and is replaced as 145 “best friend” by a new (typically female) alter. While 146 males’ “best friends” remain slightly female-biased 147 throughout their lives, women’s only eventually become 148 so during their early 50s. The two sexes eventually 149 converge on a slightly female-biased pattern at around 150 70 years of age. 151

The pattern for the “second best friend” (Fig. 1b) 152 is a partial mirror image of the pattern for the “best 153 friends”. “Second best friends” are typically same-sex, 154 reaching a sharp peak during the subjects’ early 20s 155 before falling away gradually to reverse the gender bias 156 for individuals in their late 30s. The transition is sharper 157 for women than for men: males exhibit a shallower 158 decline, and settle at an asymptotic value very close to 159 gender equality, whereas women show a striking 160 reversal to a strong male-biased peak in their late 40s 161 (and a steady decline back towards equality by the late 162 60s). We ran a similar analysis for the gender of the 163 third, fourth, and fifth best friend and since the patterns 164 are virtually identical to that in Fig. 1b (albeit with a 165 strong male-bias in older age for both sexes), we present 166 the results only in the ESM. The similarity of the plots 167 for the second, third, fourth, and fifth best friends 168 reinforce the contrast with the case for the “best friend”, 169 suggesting a more privileged status for the “best friend”. 170

In Fig. 2 we illustrate these findings in the form of 171 a network with links representing gender correlations 172 between the “best friends”. Red circles correspond to 173

female, blue circles to male subscribers, and grey circles 174 correspond to subscribers with unavailable age and 175 gender information (or subscribers of other service 176 providers). The thicknesses of the links (and the 177 number) stand for the frequencies of contact, thus 178 illustrating the emotional closeness between the pair of 179 individuals. In addition, we have used the circle sizes to 180 reflect the subscriber ages: the bigger the circle, the 181 older the subscriber. Beside gender correlations, this 182 local weighted network shows the age correlations 183 between the “best friends”. It can be seen that young 184 people prefer the “best friend” to be of opposite gender 185 and of the same age group. One can also see very 186 distinct patterns in older individuals’ communication 187 patterns, namely that a 50 year old female subscriber 188 has a young female (possibly a daughter) as the “best 189 friend” and as the “second best friend” a male of her 190 own age group (possibly her husband). What we also 191 see in this case is that the “third best friend” is typically 192 also of younger generation but male (possibly son). 193 Note the very strong opposite-gender focus of 194 relationships among 20 and 30 year olds, suggesting 195 strong pairbond focus. 196

We can see these effects more clearly by 197 considering age correlations between “best friends”. In 198 Fig. 3, we show age distributions of the “best friends” 199 for both male and female egos aged 25 and 50 years. 200 (Similar plots for the intervening age cohorts are given 201 in ESM.) On this finer scale analysis, some additional 202 patterns emerge. The distributions for friends of both 203 genders turn out to be bimodal, with one maximum at 204 around ego's own age and the other at an age difference 205 of approximately 25 years, i.e. one generation apart. The 206

Figure 2. A sample of local network between best friends. A part of the network with a gender and age correlations. Blue circles correspond to male and red circles to female subscribers. Circle sizes reflect subscriber ages: the bigger the circle, the older the subscriber. Grey circles correspond to subscribers, whose gender and age information is not available in our data set.

maxima at ego’s own age are opposite-gender biased, 207 and most likely identify a male partner for female egos 208 and vice versa. The maxima at the 25 year age 209 difference (i.e. the generation gap) show a more 210 balanced gender ratio, most likely identifying children 211 and parents, respectively, for 50-and 25-year-old egos. 212 In ESM, the progression of this split can be seen very 213 clearly in the profiles for the intervening age cohorts. 214

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Figure 3. Distributions of the “best friends” by age. The distributions of the “best friends” by age for 25 years old (a) male and (b) female egos. In c and d we show similar distributions for 50 years old male and female egos, respectively. Red circles correspond to female “best friends” while blue squares to male “best friends”.

216 3. CONCLUSION AND DISCUSSION 217

These results allow us to draw four conclusions. 218 First, women are more focused on opposite-sex 219 relationships than men are during the reproductively 220 active period of their lives, suggesting that they invest 221 more heavily in creating and maintaining pairbonds than 222 men do [17]. Second, as they age, women’s attention 223 shifts from their spouse to younger females, whom we 224 assume, on the basis of the age difference, to be their 225 daughters. This transition is relatively smooth and slow 226 for women (perhaps taking about 15 years to reach its 227 new asymptote at around age 60), and may reflect the 228 gradual arrival of successive grandchildren. Third, 229 women switch individuals around in their preference 230 rankings much more than men do, suggesting that their 231 relationships are more focussed while men’s are more 232 diffuse. Men tend to keep a steadier pattern over a 233 longer period, maintaining a preference for placing their 234 spouse in pole position across time and a striking 235 tendency to maintain a very even gender balance in the 236 second position of preference. If the latter represent 237 offspring, then it suggests a strong lack of 238 discrimination. In contrast, women tend to switch 239 individuals from one position to another in a more 240 exaggerated way, perhaps reflecting shifts in their 241

allegiances as their reproductive strategies switch more 242 explicitly from mate choice to personal reproduction to 243 (grand-)parental investment, particularly after age 40. 244 Women’s gender-biases thus tend to be stronger than 245 men’s, seemingly because their patterns of social 246 contact are strongly driven by the changes in the 247 patterns of reproductive investment across the lifespan. 248 Women’s stronger inclination toward parental and 249 grandparental investment is indicated by the fact that 250 men’s gender-biases for both best and second/third best 251 friends show much less evidence for any preference for 252 contacting children: the younger (25-year) peak for 50- 253 year-old men is half that for women and shows a more 254 even sex balance, whereas that for women is strongly 255 biased in favour of female alters (presumably, 256 daughters) (Fig. 3c,d), reflecting the striking maternal 257 grandmother investment effects previously noted in 258 demographic studies [18,19]. Finally, fourth, our results 259 provide strong evidence for the importance of female 260 matrilineal relationships in human social organisation. 261 There has been a tendency to emphasise the importance 262 of male-male relationships in an essentially patrilineal 263 form of social organisation as defining human sociality 264 [20], but our results tend to support the claim that 265 mother-daughter relationships play a particularly 266 seminal role in structuring human social relationships, 267 as has been suggested by some sociological studies [21]. 268

While, inevitably, our analyses pool together large 269 numbers of individuals, and so lose some of the richness 270 of the original data, nonetheless we have been able to 271 demonstrate striking patterns in mobile phone usage 272 data that reflect shifts in relationship preferences as a 273 function of the way the reproductive strategies of the 274 two sexes change across the lifespan. Such patterns 275 have not been noted previously, and our findings 276 suggest that there are novel opportunities for exploiting 277 large network datasets of this kind if the right kinds of 278 questions are asked. Aside from this purely 279 methodological aspect, our analyses identify striking 280 sex differences in the social and reproductive strategies 281 of the two sexes that have not been previously 282 identified. We have also been able to demonstrate a 283 marked sex difference in investment in relationships 284 during the period of pairbond formation, suggesting that 285 women invest much more heavily in pairbonds than do 286 men. Though previously suspected [17], this suggestion 287 has proved particularly difficult to test. Finally, we 288 should note that our analyses have focussed on the 289 simple presence/absence of contacts: further insights 290 into sex and age related differences in human 291 communication patterns might be gained by analysing 292 both the link directionalities or asymmetries in who 293 initiates communication and the structural and temporal 294 motifs of such directional networks. 295 ACKNOWLEDGMENTS 296

Financial support from EU’s 7th Framework 297 Program’s FET-Open to ICTeCollective project no. 298

238597 and by the Academy of Finland, the Finnish 299 Center of Excellence program, project no. 129670, and 300 TEKES (FiDiPro) are gratefully acknowledged. 301

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Electronic Supplementary Material 395

A. Dataset 396 The database consists of about 33.2 million phone service subscribers of which 6.8 397

million users are initial provider subscribers while the rest are subscribers of other 398 providers. The database contains the age and gender information about given provider 399 subscribers only. The number of male and female subscribers of known age used in the 400 analyses is NM !1.8 million and NF !1.4 million, respectively. The distributions by age 401 of male and female individuals in the sample are shown in fig. S1A. The number of males 402 exceeds the number of females for each age group. The difference between the number of 403 males and females is shown in Fig. S1A by grey colour and gives rise to overall average 404 gender g = 0.13 . 405

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Fig. S1. A: Distribution of specified provider subscribers by age. Blue and red lines correspond to males and

females, respectively. Grey lines show the difference between the number of male and female subscribers of a given age. B: Distribution of the egos’ best friends by age. Blue and red lines correspond to male and female

best friends while grey lines give the difference between them. 406

Fig. S1B shows the distribution of the egos’ best friends by age for each gender separately. 407 Grey colour in this figure corresponds to the difference between the numbers of male and 408 female best friends. The set of the best friends is very slightly female-biased with average 409 gender f = !0.01 . Here each subscriber is counted as many times as often he or she is the 410 best friend for other subscribers. For example, 23 years old female shown in fig. 2 is 411 counted twice as she is the best friend for both her possible boyfriend/husband and her 412 mother. 413 414 B. Data filtering 415

Some additional filtering of the original dataset turned out necessary because of the 416 kinds of artefact described below. Taking into account the distribution of birthdays, one 417 may state that the probability for an ego to have the best friend of his or her own age and 418 that with one-year difference should not differ significantly. Fig. S2A shows the 419 distribution of the best friends by ages for 50 year old female egos. High peaks for ego's 420 own age and gender contradict this assertion and may be caused by multiple subscriptions 421 registered for a single person. Hence, this artefact needs to be eliminated. 422

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Fig. S2. The distributions of the best friend's age for 50 years old female egos.

Panel A shows the distribution for original dataset and B shows the same distributions after filtering. 423

Different approaches to the data filtering may be applied. Our approach consists of 424 predicting the number nr (a,g) of real ego-friend pairs of equal age a and gender g . 425 Dividing nr (a,g) by the total number of such pairs n(a,g) in the database, one arrives at 426 the fraction r(a,g) = nr (a,g) / n(a,g) of real ego-friend pairs of equal age a and gender g . 427 In order to obtain reliable results for the average gender of the best friend, each ego-friend 428 pair of equal age a and gender g is taken into account with probability r(a,g) . 429

In the current study, we suppose that the number nr (a,g) of real ego-friend pairs of 430 equal age a and gender g is equal to the bigger of the number represented by pairs of 431 friends who are one year older or younger than the ego considered. In fig. S2B, we show 432 the distribution of the egos’ best friends by age for 50 years old female egos after the 433 filtering procedure is applied. The same approach is applied to the pairs of ego and second 434 best friend of equal age and gender. 435 436 C. Gender correlations in different age groups 437

Here we consider in detail how the focus of subscribers shifts between different age 438 groups (from spouses to children). For this purpose, we first have to define ego’s own age 439 group. The latter may be done considering the age distributions of the best friends (see fig. 440 3). Here one may consider the best friend to be of ego’s own age group if the ages of two 441 do not differ by more than, say, 15 years. (This is of course a rather ad hoc choice; 442 however, we ran a sensitivity analysis with alternative values without seeing any significant 443 changes). Defining ego’s age group in this way, we may consider the average gender of the 444 best friend of ego’s own age group. The results are plotted in fig. S3A. The best friend of 445 ego’s age group is almost always opposite-gender biased. 446

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(B) as a function of the ego’s age. 447

Average gender of the best friend outside ego’s own age group (fig. S3B) is less biased and 448 is usually shifted to the female direction for both male and female egos. Only for egos aged 449 between 40 and 50 is the best friend slightly male biased. Even if ego’s own age group is 450 restricted with an age difference up to 10 years, the results shown in fig. S3 do not change 451 significantly. 452

Fig. S4 shows the fraction of those best friends who belong to ego’s own age group, 453 for the male and female subscribers separately. These curves have similar shapes, 454 suggesting that both males and females change their focus from coevals to children while 455 aging. However, the positive values of the difference between male and female curves 456 (with average value ! " 0.061 ) show that males are slightly more interested in their own 457 age group than females are, while females seem to be more involved in parent-offspring 458 communication than males are. 459

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Fig. S4. Fraction of the best friends belonging to ego’s own age group (panel A), and fraction of the

second best friends belonging to ego’s own age group (panel B). Blue squares correspond to males and red circles correspond to females. Black squares show the difference between male and female curves. Positive

values of this difference (with average ! " 0.061 for the best friend and ! " 0.022 for the second best friend) show that males are slightly more interested in their own age group than females are, while females may be more involved in parent-offspring communication than males. Note that this effect is more visible

for the best friend than for the second best friend.

461 D. Different definitions of ego’s “best friend” 462

In this paper, we have so far defined the ego’s best friend on a basis of numbers of 463

calls and text messages between subscribers. Here we analyse how different definitions of 464 the ego’s best friend affect the results. In fig. S5, we consider the average gender of the best 465 friend and the second best friend if emotional closeness between subscribers is defined on a 466 basis of the number of calls only (panels A and B), total call duration (panels C and D), and 467 of the number of only text messages (panels E and F) between subscribers. The differences 468 are marginal. 469

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Ave

rage g

ender

Age of ego

C

-0.2

-0.1

0

0.1

0.2

20 30 40 50 60 70 80

Ave

rage g

ender

Age of ego

D

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

20 30 40 50 60 70 80

Ave

rage g

ender

Age of ego

E

-0.2

-0.1

0

0.1

0.2

20 30 40 50 60 70 80

Ave

rage g

ender

Age of ego

F

Fig. S5. Average gender of the best friend (left column) and the second best friend (right column) as

a function of ego’s age. Different rows correspond to different basis of the best friend definition: panels A, B (only number of calls); C, D (total call duration); E, F (number of text messages)

between subscribers. 470 E. Third, fourth, and fifth best friends 471

Fig. S6 shows average gender of the ego’s third best friend (panel A), fourth best 472 friend (panel B), and fifth best friend (panel C) as a function of ego’s age. 473

-0.2

-0.1

0

0.1

0.2

0.3

20 30 40 50 60 70 80

Ave

rag

e g

en

de

r

Age of ego

A

-0.2

-0.1

0

0.1

0.2

0.3

20 30 40 50 60 70 80

Ave

rag

e g

en

de

r

Age of ego

B

-0.2

-0.1

0

0.1

0.2

0.3

20 30 40 50 60 70 80

Ave

rag

e g

en

de

r

Age of ego

C

Fig. S6. Average gender of the ego’s third best friend (panel A), fourth best friend (panel B), and fifth best

friend (panel C) of specified age and gender. Blue squares correspond to males, red circles to females. 474 F. Age-Specific Contact Profiles 475

Contact profiles for different age and gender groups are presented in figs. S7 and S8. 476 In each figure, two (or even three) maxima as well as asymmetry in opposite gender ages 477 around ego's age are evident. Comparing the first maxima (at around ego’s own age) and 478 the second one (that with ~ 25 year difference from egos’ own age) suggest the following 479 features. If the second maximum corresponds to older individuals than ego (in the case of 480 egos aged 20 – 30 years), then the locations of male and female best friend maxima are a 481 few years shifted. In contrast, if the second maximum corresponds to alters who are 482 younger than egos (as in the case of egos aged 50 – 70 years), then the location of the male 483 and female maxima coincide for best friends. The latter may serve as additional evidence of 484 family relations among the first and second maxima best friends. 485

Figs. S9 and S10 plot the age distributions of second best friends for the two genders of 486 ego. The profiles are quite similar to those for best friends, but with striking differences in 487 the gender of the friend: for younger egos, the second best friend of similar age to ego is 488 more typically of the same gender as ego, whereas for older egos the second best friend of 489 the same age as ego is more often of the opposite gender. 490

491

492 493

0

0.02

0.04

0.06

0.08

0.1

10 20 30 40 50 60 70 80

P(a

f)


EGO: 20 YEARS OLD MALE

0

0.02

0.04

0.06

0.08

0.1

10 20 30 40 50 60 70 80

P(a

f)


EGO: 20 YEARS OLD FEMALE

0

0.02

0.04

0.06

0.08

0.1

10 20 30 40 50 60 70 80

P(a

f)



0

0.02

0.04

0.06

0.08

0.1

10 20 30 40 50 60 70 80

P(a

f)



0

0.02

0.04

0.06

0.08

10 20 30 40 50 60 70 80

P(a

f)



0

0.02

0.04

0.06

0.08

10 20 30 40 50 60 70 80

P(a

f)



Fig. S7. The distributions of the best friends by age for male and female egos of fixed age of 20, 30, and 40

years. 494

495

496 497

0

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

10 20 30 40 50 60 70 80

P(a

f)



Fig. S8. The distributions of the best friends by age for male and female egos of fixed age of 50, 60, and 70

years. 498

499 500

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

10 20 30 40 50 60 70 80

P(a

f)



0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

0.04

0.05

0.06

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

0.04

0.05

0.06

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

0.04

10 20 30 40 50 60 70 80

P(a

f)



Fig. S9. The distribution of the second best friends by age for male and female egos of different age cohorts.

501 502

503 504

0

0.01

0.02

0.03

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

10 20 30 40 50 60 70 80

P(a

f)



0

0.01

0.02

0.03

10 20 30 40 50 60 70 80

P(a

f)



Fig. S10. The distribution of the second best friends by age for male and female egos of different age cohorts. 505 506 507

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