Homophily and minority size explain perception biases in social networks

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1 Homopily and minority size explain perception biases in social networks Eun Lee,+,*, Fariba Karimi,3,+,*, Claudia Wagner,3, Hang-Hyun Jo,5,6, Markus Stromaier,7, and Mirta Galesic 8,9 Department of Matematics, University of Nort Carolina at Capel Hill, 7599, USA Department of Computational Social Science, GESIS, Cologne, 5667, Germany 3 Institute for Web Science and Tecnologies, University of Koblenz-Landau, Koblenz, Germany Asia Pacific Center for Teoretical Pysics, Poang 37673, Republic of Korea 5 Department of Pysics, Poang University of Science and Tecnology, Poang 37673, Republic of Korea 6 Department of Computer Science, Aalto University, Espoo FI-76, Finland 7 HumTec Institute, RWTH Aacen University, Germany 8 Santa Fe Institute, Santa Fe, NM 9 Harding Center for Risk Literacy, Max Planck Institute for Human Development, Berlin, Germany + Tese autors contributed equally to tis work. * Corresponding autors. fariba.karimi@gesis.org, eunfeel@ .unc.edu Abstract People s perceptions about te frequency of attributes in teir social networks sometimes sow false consensus, or overestimation of te frequency of own attributes, and sometimes false uniqueness, or underestimation of te frequency. Here we sow tat bot perception biases can emerge solely from te structural properties of social networks. Using a generative network model, we sow analytically tat perception biases depend on te level of omopily and its asymmetric nature, as well as on te size of minority group. Model predictions correspond to empirical data from a cross-cultural survey study and to numerical calculations on six real-world networks. We also sow in wat circumstances individuals can reduce teir biases by relying on perceptions of teir neigbors. Tis study advances our understanding of te impact of network structure on perception biases and offers a quantitative approac for addressing related issues in society Introduction People s perceptions of teir social worlds determine teir own personal aspirations and willingness to engage in different beaviors, ranging from voting and energy conservation 3 to ealt beavior, drinking 5 and smoking 6. Yet wen forming tese perceptions, people seldom ave an opportunity to draw representative samples from te overall social network. Instead, teir samples are constrained by te local structure of teir personal networks, wic can bias teir perception of te frequency of different attributes in te general population. For example, during te 6 U.S. presidential election Twitter users, including journalists, wo supported one candidate isolated temselves from users wo supported a different candidate 7. Members of suc insular communities often tend to overestimate te frequency of teir own attributes in te overall society. Suc social perception biases ave been studied in te literatures on false consensus 8.However, it as also been documented tat people olding a particular view sometimes tend to underestimate te frequency of tat view. Tis apparently contradictory bias as been studied in te literature on false uniqueness, pluralistic ignorance 5, 6 and majority illusion 7. It as been observed tat diverse social perception biases can be related to te structural properties of personal networks 8, 9, wic can strongly affect te samples of information tat individuals rely on wen forming teir social perceptions,. However, te impact of different network properties on social perception biases as not yet been systematically explored. Most existing explanations of tese biases instead rely on assumptions about motivational and cognitive processes, including feeling good wen oters sare one s own view, wisful tinking 3, easier recall of te reasons for aving own view 8, justifying one s undesirable beaviors by overestimating teir frequency in te society, and rational inference of population frequencies based on own attributes 5. Tese accounts can explain biases suc as false consensus but not false uniqueness. Here we sow empirically, analytically, and numerically ow a simple network model can explain social perception biases resembling bot false consensus and false uniqueness, witout furter assumptions about biased motivational or cognitive processes. Empirical results from a cross-cultural survey sow tat structural network properties, namely, level of omopily and minority-group size, influence people s social perception biases. Analytical results from a generative network model wit

2 tunable omopily and minority-group size align well wit te empirical findings. Numerical investigations sow tat model predictions are consistent wit sample biases tat could occur in six empirical networks, and point to te importance of an additional structural property, asymmetric omopily. We also sow wen perception biases can be reduced by aggregating one s own perceptions wit te perceptions of one s neigbors. We discuss te implications of tese results for te understanding of te nature of uman social cognition and related social penomena Results Defining social perception biases We focus on perceptions of te prevalence of binary attributes (e.g., smoking, believing in a god, or donating to carity) in te overall network. We define social perception biases on te individual and te group level. Unless oterwise stated, we use P indv. and P group to denote individual- and group-level perception bias. If necessary, we add a superscript a for te minority and b for te majority group. On te individual level, we assume tat individuals perceptions are based on te prevalence of an attribute in teir personal network (direct neigborood). We define individual i s bias in perception of te fraction of individuals wit a given attribute in te overall network as follows: 6 P indv. = f a j Λi x j k i, () were k i is te degree of individual i, tat is, te size of i s personal network, Λ i is a set of i s neigbors, x j is te attribute of individual i s neigbor j (x j = for a minority attribute and for a majority attribute), and f a is te true fraction of minority nodes in te overall network. Te group-level perception bias is defined by aggregating perception biases of all individuals in a minority or a majority group: 67 P group = f a i Ng j Λi x j i Ng k i, () were N g is te set of individuals in a group g, wic can be a minority or a majority group. Tis definition is comparable to tose used in studies of te friendsip paradox 6, 7 and enables us to derive analytic predictions at te group level using a mean-field approac. Te minimum value of te perception bias is and te maximum value is / f a (see Metod). A perception-bias value below indicates an underestimation of te minority-group size, and a value above indicates an overestimation. If te value equals a group or an individual perfectly perceives te fraction of a minority attribute in te overall network. As an example, Fig. illustrates ow we define te perception bias at te individual level and group level for a igomopily (omopilic) network and a low-omopily (eteropilic) network. Here, te color of an individual node depicts its group membersip. Orange nodes belong to te minority and blue nodes to te majority. We focus on te central individual i wo is in te majority in bot networks. Tis individual estimates te size of te minority group on te basis of te fraction of orange nodes in is personal network (dased circle). In te omopilic network (Fig. a), is individual-level perception bias is (/6)/(/3) =.5, wic means tat e underestimates te size of te minority group by a factor of.5. Consequently, e overestimates te size of is own, majority group in te overall network. In te eteropilic network (Fig. b), te perception bias of individual i is (/6)/(/3) =, implying tat e overestimates te size of te minority group by a factor of. At te group level, te majority group (blue) perceives te size of te minority group to be /8 in te omopilic network and /8 in te eteropilic network. Terefore, te majority group underestimates te size of te minority group by a factor of (/8)/(/3) =.67 in te omopilic network and overestimates it by a factor of (/8)/(/3) =.67 in te eteropilic network. Survey of social perception biases To gain insigt into te role of omopily and minority-group size in social perception biases, we conducted a survey wit N = participants from te United States and N = 99 participants from Germany, recruited from Amazon s Mecanical Turk, and N = Sout Korean participants recruited from te survey platform Tillion Panel. For American and German participants, we asked questions about attributes, listed in Table S. Te questions were taken from existing national surveys /6

3 tat provided objective frequencies of different attributes in te general population of eac country. For Korean participants, we asked questions about seven of tese attributes for wic we ad objective frequencies from national surveys (see Table S). Participants answered tree groups of questions. First, tey answered questions about teir own attributes (e.g., smoking). Second, tey estimated te frequency of people wit eac attribute in teir personal networks, defined as all adults you were in personal, face-to-face contact wit at least twice tis year. We used tese answers to calculate te omopily in teir personal networks (see Metod for more details). For example, if a participant was a smoker and 7% of er social contacts were smokers, te probability of a friendsip link between tis participant to a smoker is 7%. We used tis information to estimate te omopily parameter () for eac individual s personal network using Eq.. Homopily can vary from (complete eteropily) to (complete omopily). Tird, participants estimated te frequency of people wit a particular attribute in te general population of teir country. We compared tese estimates wit te results from national surveys to calculate teir social perception bias (see Metod). Te perception bias was calculated as a ratio of an individual s estimate and te objective frequency from national surveys. For example, if a participant reported tat se believes tat 6% of te population smoke tobacco wereas te national survey suggested tat % do so, te perception bias was 6/ =.5. For eac country, we analyzed perception biases separately for attributes tat in tat country are objectively found to be eld by a small ( f a <.), medium (. f a <.), or large (. f a <.5) fraction of te minority group. For example, te attribute not aving money for food is eld by a small minority in all tree countries (Table S). In contrast, te attribute donating to carity is eld by a large minority in Germany and a medium minority in Sout Korea. In te United States, tis attribute is eld by a majority, and its reverse, not donating to carity, is eld by a large minority. Fig. sows te survey results for te United States and Germany (Fig. S for te Sout Korea), wit a different country in eac row. Perception bias for te size of te minority group is sown separately for participants wo belonged to te minority (left column) and majority (rigt column) groups. Te value of te perception bias indicates ow accurately eac participant (eac point in te plot) perceived te size of te minority group to be in te overall population. A perception bias of means perfect accuracy, values above indicate overestimation of te minority-group size, and values below indicate underestimation of te minority-group size. We observed clear effects of te objective fraction of te minority group in te overall population ( f a ) and of omopily of personal networks () on perception biases. As minority-group size in te overall population decreased, te size of te perception bias increased. Moreover, wen omopily in personal networks was large ( >.5), minority participants overestimated and majority participants underestimated te size of te minority, resembling false consensus. In contrast, for low levels of omopily in personal networks ( <.5), we observed a muc smaller false-consensus or even false-uniqueness tendencies for bot minority and majority participants. Similar relationsips between perception biases, omopily, and minority-group size were observed in all tree countries. Te survey results provide evidence tat omopily of personal networks and minority-group size partially explain perception biases regardless of cultural differences. Tey stress te necessity of considering te social structure in wic an individual is embedded to understand ow social perception biases are formed. In te next section we terefore explore weter tese biases can be explained by a simple generative network model Generative network model wit tunable omopily and minority-group size To systematically study te relation between perception biases and network structure, we developed a network model tat allows us to create scale-free networks wit tunable omopily and minority-group sizes 8. Tis network model is a variation of te Barabási Albert preferential attacment model wit te addition of a omopily parameter (we call tis model BA-omopily). Te probability of an attacment of a newly arriving individual (node) w to an existing node v (φ wv ) is proportional to te node s degree (k v ) and te omopily between te two nodes ( wv ); tus φ wv wv k v. Tus, te degree and te omopily parameter regulate te probability of connection between individuals wo sare te same attribute. Te value of omopily ranges from to. If omopily is low ( <.5), nodes ave a tendency to connect to nodes wit opposite attribute values. As te omopily parameter increases above.5, te probability of connection between nodes wit te same attribute increases. In te case of extreme omopily ( = ), nodes strictly connect to oter nodes wit te same attribute value and terefore two separate communities emerge. One advantage of using te BA-omopily model is tat it generates networks wit te scale-free degree distributions observed in many large-scale social networks. Anoter advantage is tat it generates networks in wic omopily can be symmetric or asymmetric. Given nodes wit two attributes, a and b, wen omopily is symmetric te tendency to connect to nodes wit te same attribute is te same for bot groups, aa = bb =. In tis case one parameter is sufficient to generate te network. In te asymmetric omopily case, two omopily parameters are needed to regulate te connectivity probabilities for eac group separately, tus aa bb. Fig. 3 depicts analytically derived biases in perceptions of minority-group size among individuals wo temselves belong to a minority (Fig. 3a) or a majority (Fig. 3b) group, as a function of te true fraction of te minority in te overall network 3/6

4 ( f a ) and te omopily of individuals personal networks (). Te solid lines sow te analytic results (see Metod) and te circles are numerical results obtained from te BA-omopily model. In eteropilic networks ( <.5), perception biases resemble false uniqueness. Te minority underestimates its own size, and te majority overestimates te size of te minority, te more so te smaller te minority group (smaller f a ). In omopilic networks (.5 < ), perception biases resemble false consensus. Te minority overestimates its own size (te more so te smaller te minority group), and te majority underestimates te size of te minority. Sligt asymmetries between biases expected for minority and majority groups (see insets in Fig. 3) are due to te disproportionate number of links for two groups, affecting te results of Eq. 7. Tese analytic derivations can elp us describe te functional form of te biases observed in te survey (Fig. ). As sown in Eq. 8, te minority s perception bias () a is proportional to te density of links between minorities (p aa ), wic increases wit te omopily between minority nodes, tat is, aa. Similarly, te majority s perception bias () b is proportional to te density of intergroup links (p ab ), wic decreases as te omopily ( aa and bb ) increases. In addition, te relative sizes of minority and majority groups influence te growt rate of links for eac group according to Eq. 9 so tat perception biases can increase (or decrease) nonlinearly wit group size (see te Supplementary Materials). For instance, in te extreme omopily case wit =, one gets p aa = p bb =, wile p ab = p ba =, leading to te minority s group-level perception bias of / f a. In sum, te proposed BA-omopily model and its analytic derivations facilitate systematic understanding of ow network structure affects perception biases. Wile we find general agreement between te survey results and our BA-omopily model calculations, tere are some differences tat call for more detailed investigation in te future. One main difference is tat in survey results (Fig. ) we observed perception bias > in some cases wen Fig. 3 predicts it to be <. Specifically, tis tends to appen for small minority sizes, wen <.5 for minority and wen >.5 for majority participants. A possible explanation tat is in line wit previous studies in social cognition is tat people do not observe and report attribute frequencies in teir samples (ere, teir personal networks) completely accurately, but wit some random noise. Tis does not matter on average wen minority size is relatively large as errors of over- and underestimation cancel out. But for small minority-group size te estimate cannot be lower tan, meaning tat te former errors (overestimated - true sample frequency) will be larger tan te latter errors (true - underestimated sample frequency) and will not cancel out. Hence, people s estimates of te frequency of attributes in teir samples will be overestimated for small minority groups, wic is wat we observed in survey results Social perception biases in real-world networks Te BA-omopily model offers a very simple representation of real-world networks. To examine possible social perception biases in te real world, we studied six empirical networks wit various ranges of omopily and minority-group sizes. Te network caracteristics of te data sets are presented in Table. Te detailed descriptions of te data sets and references can be found in te Metod section. Tese empirical networks ave different structural caracteristics and sow different levels of omopily or eteropily wit respect to one specific attribute (see Supplementary Materials). In five of te networks tis attribute is gender (female or male), wile in one - te American Pysical Society (APS) network - te attribute is weter a paper belongs to te field of classical statistical mecanics or quantum statistical mecanics. To estimate omopily, we first assumed tat omopily is symmetric in all networks. Te symmetric omopily is equivalent to Newman s assortativity measure (q), wic is te Pearson coefficient of correlation between pairs of linked nodes according to some attribute (e.g., race 9 ). Te Newman assortativity measure corresponds directly to te omopily parameter in our model wen adjusted for te scale. In our model = means complete eteropily (q = ), =.5 indicates no relationsip between structure and attributes (q = ), and = indicates complete omopily (q = ; see te Supplementary Materials). In reality, owever, te tendency of groups to connect to oter groups can be asymmetric. For example, it as been observed tat in scientific collaborations, omopily among women is stronger tan omopily among men 3. Given te relationsip between number of edges tat run between nodes of te same group and omopily in Eqs. and, we can estimate te asymmetric omopily, wic differs for te minority ( aa ) and te majority ( bb ) group (see Metod). As we describe below, it turns out tat asymmetric omopily as an important impact on te predictability of perception bias in empirical networks. We used te measured omopily and minority-group size in te empirical networks (Table ) to generate syntetic networks wit similar caracteristics to tose of te six empirical social networks. Tis enabled us to compare perception biases in empirical and syntetic networks and gain insigts about te impact of omopily and minority-group size differences on possible individual- and group-level perception biases. Fig. sows group-level perception biases in te empirical networks tat could occur if people s perceptions were based solely on te samples of information from teir personal networks. Because furter cognitive or motivational processes could affect te final perceptions, tese estimates can be taken as a baseline level of biases tat could occur witout any additional psycological assumptions. Te overall trends sown in Fig. are in agreement wit te results obtained from te survey and /6

5 from te syntetic networks. In eteropilic networks, te minority group is likely to underestimate its own group size and te majority group is likely to overestimate te size of te minority. Conversely, in te omopilic networks, te minority group is likely to overestimate its own size and te majority group to underestimate te size of te minority. We can compare perception biases estimated directly from empirical networks (crosses in Fig. ) wit tose estimated from syntetic networks wit similar omopily and minority-group size. In Fig., triangles correspond to networks wit symmetric omopily and squares to networks wit asymmetric omopily. Altoug symmetric omopily traces empirically observed perception biases in most instances, it fails to capture te biases in te GitHub network, especially for te minority group. Tis network exibits a iger level of asymmetric omopily compared to oter networks (see Table ). Wen perception biases are estimated from a syntetic model tat assumes asymmetric omopily, tey approximate perception biases of bot te minority and te majority groups very well in all networks. Tis suggests tat asymmetric omopily plays an important role in saping possible perception biases. It is known tat influential nodes in networks, usually identified by teir ig degree, can affect processes in networks suc as opinion dynamics 3, social learning 3, and wisdom of crowds 33. To evaluate te impact of degree on saping perception biases, we plotted individual perception biases, P indv., versus individual degree in Fig. S in supplementary. Te distribution of individual perception biases estimated from te BA-omopily model mostly corresponds to te empirically estimated distribution. In addition, nodes wit low degrees display a iger variation in perception biases compared to nodes wit ig degrees. Te model does not explain all te variation observed in te empirical networks. Tis can be due to incomplete observations of all social contacts in real networks or to oter processes tat we did not consider in generating te model. However, te model can still predict te trend we observed in te empirical data, wic would not be predicted assuming random connectivity among individuals. Tis analysis enables us to understand te impact of influential individuals on perception biases exibited by oter individuals in te network. Reducing social perception biases We investigated to wat extent and under wat structural conditions individuals can reduce teir perception bias by considering te perceptions of teir neigbors. We built on DeGroot s weigted belief formalization by aggregating an individual s own perception (ego) wit te averaged perceptions of te individual s direct neigbors (-op) 3. For simplicity, we assumed symmetric omopily in te BA-omopily model (te results for asymmetric omopily are in te Supplementary Materials). Fig. 5 sows a comparison of te average perception bias ( P indv. ) for individuals wo belong to te (a) minority and (b) majority and te weigted averages of teir own perceptions and tose of teir -op neigbors. Te minority-group size is fixed to.. Te results sow tat taking into account perceptions of -op neigbors improves estimates of individuals in eteropilic networks (blue triangles are closer to te gray dased line compared to orange circles in te log scale). Te improvement in te perception is te result of nodes being more likely to be exposed to neigbors wit opposing attributes. In omopilic networks, including neigbors perceptions does not lead to a significant improvement because individuals are exposed to neigbors wit similar attributes to teir own. Tis suggests tat in omopilic networks, individuals cannot improve teir perceptions by consulting teir peers because tey are similar to tem and do not add new information tat migt not increase te accuracy of teir estimates. However, in eteropilic networks, individuals benefit from considering teir neigbors perceptions because tey are not exposed only to oter like-minded individuals but to more diverse perceptions. Wile te overall trend is not surprising, our results reveal ow te accuracy of tese estimates canges as a function of omopily. Discussion Te way people perceive teir social networks influences teir personal beliefs and beaviors and sapes teir collective dynamics. However, many studies ave documented biases in tese social perceptions, including bot false consensus and false uniqueness. Here we investigated to wat extent bot of tese biases can be explained merely by te structure of te social networks in wic individuals are embedded, witout any assumptions about biased cognitive or motivational processes. Using empirical survey data, analytical investigation, and numerical simulations, we sow tat structural properties of personal networks strongly affect te samples people draw from te overall population. We find tat biased samples alone can lead to apparently contradictory social perception biases suc as false consensus and false uniqueness. Wile cognitive and motivational processes undoubtedly play an important role in te formation of social perceptions, our analyses establis a baseline level of biases tat can occur witout assuming biased information processing, Our results suggest tat omopily impacts te accuracy of te estimates of individuals in bot minority and majority groups. In omopilic networks, minority group tends to overestimate its own size, and majority group tends to underestimate te size of te minority. In contrast, in eteropilic networks, minority group tends to underestimate its own size, and majority 5/6

6 group overestimates te size of te minority. In oter words, wen omopily is ig, bot minority and majority groups tend to sow social perception biases similar to false consensus, wereas wen it is low, bot group sow biases similar to false uniqueness. We furter sow tat te relative sizes of te majority and minority groups influence perception biases. Specifically, te smaller te size of te minority, te iger te false consensus among members of te minority group and false uniqueness among members of te majority group. To explain te underlying structural mecanisms of perception biases in social networks, we developed a generative network model wit tunable omopily and minority-group size. We find tat predictions from tis teoretical model correspond to empirical observations well, especially wen we assume asymmetric rater tan symmetric omopily. In addition, we sow tat perception biases can be reduced by aggregating individuals perceptions wit tose of teir direct neigbors, toug only in eteropilic networks. In omopilic networks, tese socially informed estimates do not lead to more accurate perceptions due to similarity of te nodes to teir neigbors. Most previous explanations of perception biases in social psycology did not include precise quantitative models of social environments. Many of tem did not explicitly include te structure of social networks at all. Tose tat did, suc as selective-exposure explanations, typically involved qualitative statements tat people tend to socialize wit similar oters 8, 8. Te network model described in te present paper enabled a toroug quantitative investigation of ow different levels of omopily, its asymmetric nature, and different relative sizes of majority and minority groups, affect social perception biases. Tis investigation did not include a quantitative specification of te cognitive processes underlying people s sampling from teir personal networks. Suc specifications 9,, 35, 37 could be combined wit te network model described ere. Besides providing a teoretical account of perception biases, our results ave practical implications for understanding real-world social penomena. Given te importance of omopilic interactions in many aspects of social life ranging from ealt-related beavior 39 to group performance and social identity, it is crucial to consider obstacles to improving social perceptions. Perceptions of te frequency of different beliefs and beaviors in te overall population influence people s beliefs about wat is normal and sape teir own aspirations 39,, 3. Wen people overestimate te prevalence of teir own attributes in te overall population, tey will be more likely to tink tat tey are in line wit social norms, and consequently, less likely to cange. We found tat small minorities wit ig omopily are especially likely to overestimate teir actual frequency in te overall network. If suc committed minorities become resistant to cange, tey can eventually influence te wole network, 5, and wen suc minorities ave erroneous views, te wole society could be worse off. Our results furter suggest tat a possible way to correct biases is to promote more communication wit and reliance on neigbors perceptions. However, tis can be useful only in conjunction wit promoting more diversity in people s personal networks. Promoting more communication in omopilic networks does not improve perception biases. Tis study is not witout limitations. One strong assumption in our metodology is tat one s perception is based solely on information sampled from one s direct neigborood. In te real world, individuals can also rely on oter sources suc as news reports, polls, and general education. In addition, we observe differences between te results of our survey and numerical simulations tat call for future investigation on te impact of minority-group size on perceptions. In particular, experimental group studies could lead to a better understanding of ow different network properties affect perceptions. In sum, tis study sows tat bot over-and under-estimation of te frequency of own view can be explained by different levels of omopily, asymmetric nature of omopily, and size of te minority group. Integration and quantification of te biases provide a rater compreensive picture of te baseline level of uman perception biases. We ope tat tis paper offers insigts into ow to measure and reduce perception gaps between different groups and fuels more work on understanding te impact of network structure on individual and group perceptions of our social worlds Metod BA-omopily model To gain insigt into ow network structure affects perception biases, we developed a network model tat allows us to create scale-free networks wit tunable omopily and minority-group size 8. Tis network model is a variation of te Barabási Albert model wit te addition of omopily parameter. In tis model, te probability tat a newly introduced node w connects to an existing node v is denoted by φ wv and it is proportional to te product of te degree of node v, k v, and te omopily between w and v as follows: 96 φ wv = wv k v v {G},v w wv k v. (3) 6/6

7 Here, wv is te probability of connection between nodes v and w. Tis is an intrinsic value tat depends on te group membersip of v and w. {G} is a set of nodes in a grap G. Before constructing te network, we specify two initial conditions: (i) te size of te minority group and (ii) te omopily parameter tat regulates te probability of a connection between minority and minority individuals, majority and majority individuals, minority and majority individuals, and majority and minority individuals. Eac arrival node continues te link formation process until it finds m nodes to connect to. If it fails to do so, for example, in an extreme omopily condition, te node remains in te network as an isolated node. Te parameter m guarantees te lower bound of degree and in our model is set as. Altoug tis parameter is fixed for eac node, te stocasticity of te model ensures te eterogeneity of te degree distribution Analytic derivation for group-level perception bias Let us refer to te minority as a and te majority as b. K a (t) and K b (t) are te total number of degrees for eac group of te minority and te majority at time t, respectively. At eac time step, one node arrives and connects wit m existing nodes in te network. Terefore, te total degree of te growing network at time t is K(t) = K a (t) + K b (t) = mt. In tis model, te degree growt is linear for bot groups. Denoting C as te minority s degree growt factor, we ave 3 K a (t) = Cmt, K b (t) = ( C)mt. () 3 33 Te probability of a connection between two minority nodes is te product of teir degree and omopily: p aa = aa K a (t) aa K a (t) + ab K b (t) = aa C aa C + ab ( C), (5) were aa is te omopily between minority nodes, and ab is te omopily between a minority and a majority in a relation of aa. Te connection probability from a minority to a majority is te complement of p aa as p ab = ab K b (t) aa K a (t) + ab K b (t) = ab ( C) aa C + ab ( C). (6) Similar relationsips can be found for te connection probability of majority to majority and majority to minority. To calculate te group perception bias (P group ), one needs to consider te number of inter- and intragroup edges as expressed in te definition of Eq.. Tus, te group perception bias for te minority and majority is as follows: P a group = f a M aa M aa + (M ab + M ba ), Pb group = f a M ab + M ba M bb + (M ab + M ba ). (7) Here, M represent te number of edges between different groups. For example, M aa is te number of edges between minority nodes. Note tat we distinguis number of edges between minority and majority M ab and between majority and minority M ba. Tese values are equivalent wen omopily is symmetric but tey are unequal wen te omopily is asymmetric. Since eac M as a relation as a product of te total number of edges and te edge probability, suc as M aa = mn a p aa, we can reduce Eq. 7 to te following equations: P a group = f a p aa p aa + p ab + (N b /N a )p ba, P b group = f a (N a /N b )p ab + p ba p bb + (N a /N b )p ab + p ba, (8) were N a and N b represent te number of nodes in eac group. Te analytic derivations are intuitive and well explained by te numerical results (solid lines in Fig. 3). For example, wen f a =.5 in extreme omopily ( =.) wit te degree growt C = (a symmetric omopily condition), a = from Eq. 8, and it matces well wit te numerical result in Fig. 3a. Note tat te growt parameter C is a polynomial function and its relation to omopily is sown in te Supplementary Materials. 7/6

8 Measuring omopily in empirical networks From te linear degree growt sown in Eq. S3, we can derive te relation between te degree growt C and te inter- and intralink probabilities p aa, p ab, and p bb (see te Supplementary Materials). Tus, 335 C = f a ( + p aa ) + f b p ba. (9) In empirical networks we know te edge density for te minority (r aa = M aa /M), between groups (r ab = M ab /M), and for te majority (r bb = M bb /M). Tus, te probability of connection witin a group can be written as r aa = f a p aa and r bb = f b p bb. Connection probabilities are defined as follows: p aa = aa C aa C + ab ( C), p bb ( C) bb = bb ( C) + ba C. () From Eq. and te relation between r aa and p aa (or r bb and p bb ), we can derive te empirical omopily by using edge density r aa,r bb as follows: aa = r aa ( C) f a C + r aa ( C), bb = r bb C f b ( C) r bb ( C). () Tis calculations allow to estimate te omopily from te empirical networks assuming tat te BA-omopily model is a valid model of a social network. Te omopily can be symmetric or asymmetric in its nature. In order to understand te role of asymmetricit omopily in perception biases, we first assume all empirical networks ave symmetric omopily. We approximate symmetric omopily by projecting te omopily to Newman s assortativity measure (q) (see Supplementary Materials and Fig. S). Empirical networks Te first network, Brazilian network, captures sexual contact between sex workers and sex buyers 6. Te network consists of 6,73 nodes and 39, edges. Tere are,6 sex workers and 6,6 sex buyers (minority-group size f a =.). In tis network, no edges among members of te same group exist resulting in te Newman s assortativity (q = ), and consequently, te network is purely eteropilic ( = ). Te second network is an online Swedis dating network from PussOKram.com (POK) 7. Tis network contains 9,3 nodes wit strong eteropily ( =.7,q =.65). Given te ig bipartivity of te network, we are able to infer te group of nodes using te max-cut greedy algoritm. Te results are in good agreement wit te bipartivity reported 8. Since te group definition is arbitrary, we label te nodes based on teir relative group size as minority gender and majority gender. Here, te fraction of te minority in te network is.. Te tird network is a Facebook network of a university in te United States (USF5) 9. Te network is composed of 6,53 nodes and includes information about individuals gender. In tis network male students are in te minority, occupying % of te network, and te network exibits a small eteropily 9 (q =.6, =.8). Te fourt network is extracted from te collaborative programming environment GitHub. Te network is a snapsot of te community (extracted August, 5) tat includes information about te first name and family name of te programmers. We used te first name and family name to infer te gender of te programmers 5. After we removed ambiguous names, te network consisted of,338 men and 7,33 women. Here, women belong to te minority group and represent only about 6% of te population. Te network displays a moderately symmetric gender omopily of.53 (q =.7). Te fift network depicts scientific collaborations in computer science and is extracted from Digital Bibliograpy & Library Project s website (DBLP) 5. We used a new metod tat combines names and images to infer te gender of te scientists wit ig accuracy 5. We used a -year snapsot for te network. After we filtered out ambiguous names, te resulting network included 8, scientists and 75,6 edges (paper coautorsips) wit 63,356 female scientists and 6,8 male scientists. Tis network sows a moderate level of symmetric omopily ( =.55 and q =.). Te last network is a scientific citation network of te American Pysical Society (APS). Citation networks depict te extent of attention to communities in different scientific fields. We used te Pysics and Astronomy Classification Sceme (PACS) identifier to select papers on te same topics. Here, we cose statistical pysics, termodynamics, and nonlinear dynamical systems subfields (PACS = 5). Witin a specific subfield, tere are many subtopics tat form communities of various sizes. To make te data comparable wit our model, we cose two relevant subtopics, namely, Classical Statistical Mecanics (CSM) and Quantum Statistical Mecanics (QSM). Te resulting network consists of,853 scientific papers and 3,67 citation links. Among nodes, 696 are in te minority and,57 in te majority. Here, te minority group in tese two subtopics is CSM ( f a =.37). Tis network sows te igest omopily compared to te oter empirical data sets ( =.9 and q =.87). 8/6

9 Survey study For all countries (te Unite State, Germany, and Sout Korea), we restricted participant age to at least 8 years. Regarding gender, 6.% of te U.S. participants, 85.9% of te German participants, and 5.% of te Korean participants were male. Te age distribution in te United States was 8 3 years: 3.6%, 3 years: 36.6%, 5 years: 9.8%, 5+ years: 9.%; in Germany it was 8 3 years: 6.%, 3 years: 6.3%, 5 years: 6.%, 5+ years: 6.3%; and in Sout Korea it was 8 3 years: 6.%, 3 years: 6.%, 5 years:.%, 5+ years:.%. Participants were asked questions about teir own attributes, te frequency of tese attributes in teir personal networks, and teir frequency in te overall population of teir country. Question texts and objective sizes of minority and majority groups in te overall populations were taken from publicly available results of large national surveys conducted in eac country. Details are provided in Table S. U.S. and German participants were asked about attributes and Korean participants about seven of tose attributes for wic we could find objective population data. We estimated te omopily of participants personal networks on te basis of teir reports of te size of minority and majority groups in teir social circles and following an approac similar to tat of Coleman 5. Eac survey participant reported te fraction of is or er personal network (or social circle) wo ave a specific attribute. For example, a participant wo does not smoke migt ave estimated tat 8% of er social circle are nonsmokers. We used tis fraction to calculate te probability tat any two nonsmokers in er social circle are connected. As a complementary relation of connection between attributes, we furtermore used te fraction of smokers in er social circle % to calculate te probability tat any nonsmoker and smoker are connected. Tese probabilities are equivalent to p aa or p bb, and teir complementaries correspond to p ab or p ba in te BA-omopily model. Using Eqs. 9 and we can calculate te omopily aa (or bb ) of eac participant s personal network. In addition, we can evaluate ab and ba using te relations ab = aa and ba = bb. To study te effect of minority-group size, we analyzed results separately for attributes for wic minority group size in a particular country was small ( f a <.), medium (. f a <.), and large (. f a <.5). For example, small minority attributes in te U.S. are no money for food, experienced teft, and smoking, because te objective frequency of tese attributes in te general U.S. population is smaller tan. (see Table S). We measured participants individual perception bias by dividing teir estimate of minority-group size in te general population by te objective minority-group size obtained from national surveys, according to Eq References. Cialdini, R. B. & Trost, M. R. Social influence: Social norms, conformity and compliance. In Te Handbook of Social Psycology, Vol., 5 9 (McGraw-Hill, Boston, MA, 998).. Bond, R. M. et al. A 6-million-person experiment in social influence and political mobilization. Nature 89, (). 3. Allcott, H. Social norms and energy conservation. J. Public Econ. 95, 8 95 ().. Centola, D. Te spread of beavior in an online social network experiment. Science 39, 9 97 (). 5. Borsari, B. & Carey, K. B. Descriptive and injunctive norms in college drinking: a meta-analytic integration. J. Stud. Alcool 6, 33 3 (3). 6. Botvin, G. J., Botvin, E. M., Baker, E., Dusenbury, L. & Goldberg, C. J. Te false consensus effect: Predicting adolescents tobacco use from normative expectations. Psycol. Rep. 7, 7 78 (99). 7. Journalists and Trump voters live in separate online bubbles. ttps://news.vice.com/story/ journalists-and-trump-voters-live-in-separate-online-bubbles-mit-analysis-sows Ross, L., Greene, D. & House, P. Te false consensus effect : An egocentric bias in social perception and attribution processes. J. Exp. Soc. Psycol. 3, 79 3 (977). 9. Mullen, B. et al. Te false consensus effect: A meta-analysis of 5 ypotesis tests. Journal of Experimental Social Psycology, 6 83 (985).. Krueger, J. & Clement, R. W. Te truly false consensus effect: An ineradicable and egocentric bias in social perception. J. Pers. Soc. Psycol. 67, (99).. Krueger, J. From social projection to social beaviour. Eur. Rev. Soc. Psycol. 8, 35 (7).. Frable, D. E. Being and feeling unique: Statistical deviance and psycological marginality. J. Pers. 6, 85 (993). 9/6

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