homophily and the glass ceiling effect in social networkshomophily and the glass ceiling effect in...
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Homophily and the Glass Ceiling Effect in Social Networks
Chen Avin ∗†
Ben Gurion University of theNegev, Israel
Barbara Keller ‡ETH Zurich, Switzerland
Zvi Lotker ∗Ben Gurion University of the
Negev, [email protected]
Claire MathieuCNRS, École Normale
Supérieure, [email protected]
David Peleg ∗
Weizmann Institute, [email protected]
Yvonne-Anne PignoletABB Corporate, Switzerland
ABSTRACTThe glass ceiling effect has been defined in a recent US Fed-eral Commission report as “the unseen, yet unbreakable bar-rier that keeps minorities and women from rising to the up-per rungs of the corporate ladder, regardless of their qualifi-cations or achievements”. It is well documented that manysocieties and organizations exhibit a glass ceiling. In thispaper we formally define and study the glass ceiling effect insocial networks and propose a natural mathematical model,called the biased preferential attachment model, that par-tially explains the causes of the glass ceiling effect. Thismodel consists of a network composed of two types of ver-tices, representing two sub-populations, and accommodatesthree well known social phenomena: (i) the “rich get richer”mechanism, (ii) a minority-majority partition, and (iii) ho-mophily. We prove that our model exhibits a strong mo-ment glass ceiling effect and that all three conditions arenecessary, i.e., removing any one of them will prevent theappearance of a glass ceiling effect. Additionally, we presentempirical evidence taken from a mentor-student network ofresearchers (derived from the DBLP database) that exhibitsboth a glass ceiling effect and the above three phenomena.
Categories and Subject DescriptorsG.2.2 [Mathematics of Computing]: DISCRETE MATH-EMATICS—Graph Theory ; J.4 [Computer Applications]:SOCIAL AND BEHAVIORAL SCIENCES
Keywordssocial networks; homophily; glass ceiling
∗Supported in part by the Israel Science Foundation (grant1549/13).†Part of this work was done while the author was a longterm visitor at ICERM, Brown university.‡Part of this work was done while the author was a visitingstudent at the Weizmann Institute.
Permission to make digital or hard copies of all or part of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full cita-tion on the first page. Copyrights for components of this work owned by others thanACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re-publish, to post on servers or to redistribute to lists, requires prior specific permissionand/or a fee. Request permissions from [email protected]’15, January 11–13, 2015, Rehovot, Israel.Copyright c© 2015 ACM 978-1-4503-3333-7/15/01 ...$15.00.http://dx.doi.org/10.1145/2688073.2688097.
1. INTRODUCTIONAttaining equality of opportunity is a fundamental value in
democratic societies, therefore existing inequalities presentus with a major concern. A particularly sore example isthat many highly-qualified women and members of minoritygroups are unable to realize their full potential in society(and specifically in the workforce) due to a phenomenoncommonly referred to as the glass ceiling, a powerful visualimage for an invisible barrier blocking women and minoritiesfrom advancing past middle management levels [20]. Thisconcern was raised in a recent US Federal commission report[18]:
The “glass ceiling”... is the unseen, yet unbreak-able barrier that keeps minorities and women fromrising to the upper rungs of the corporate ladder,regardless of their qualifications or achievements.
The existence of the glass ceiling effect is well documented[8, 16, 30]. In academia, for example, gender disparitieshave been observed in the number of professors [34], earn-ings [13, 34, 40] funding [29] and patents [10]. A recentstudy [26] analyzed gender differences in research output, re-search impact and collaborations based on Thomson ReutersWeb of Science databases. When prominent author posi-tions were analyzed by sole authorship, first-authorship andlast-authorship, it was discovered that papers with womenin those leading roles were less frequently cited. The ques-tion we focus on in this article concerns the causes of thisphenomenon. What are the invisible mechanisms that com-bine to create the glass ceiling effect, and in particular, whatis the role of the social network in creating this effect? Manypapers discuss possible causes of the glass ceiling effect andpotential solutions to it, e.g., [9, 15, 24], but to the bestof our knowledge, the present work is the first attempt tostudy it in the context of the social network structure and topropose a mathematical model capturing this phenomenon.
The paper’s main contributions are the following. (1) Wepropose a model for bi-populated social networks extendingthe classical preferential attachment model [2], and augmentit by including two additional basic phenomena, namely, aminority-majority partition, and homophily. (2) We proposea formal definition for the glass ceiling effect in social net-works. (3) We rigorously analyze this extended model andestablish its suitability as a possible mechanism for the emer-gence of a glass ceiling effect. We also show that omittingany one of the three ingredients of our model prevents theoccurrence of a glass ceiling effect. (4) We present empirical
evidence for a network exhibiting preferential attachment,minority-majority partition, homophily, and a glass ceilingeffect.
In order to talk about the glass ceiling effect we have toagree on a measure of success in a social network. Followingthe traditional approach that sees network edges as the “so-cial capital” of the network, we define successful members ofa social network to be high degree vertices, namely, verticesthat maintain a large number of connections, correspond-ing to high influence. We base our model on a bi-populatednetwork augmented by three well-accepted observations onhuman behavior, namely (i) the“rich get richer”mechanism,(ii) minority-majority partition (with a slower growth rateof the minority group in the network), and (iii) homophily(affinity towards those similar to oneself). The main resultof the paper is that under these three simple and standardassumptions the glass ceiling effect naturally arises in so-cial networks. Let us first briefly describe these three socialphenomena.
The “rich get richer” mechanism. This mechanismdescribes and explains the process of wealth concentration.It follows the basic idea that newly created wealth is dis-tributed among members of society in proportion to theamount they have already amassed. In our setting, wherethe degree of the vertex captures its level of social wealth,this mechanism predicts that people may try to connectmore often to people who already have many connections,either in order to profit from their social wealth or becausethey are more visible in the network.
Minority-majority partition. Many social groups ex-hibit unequal proportions of men and women. Certain oc-cupations, such as construction, law enforcement, politicsand computer science, tend to have a higher proportion ofmen. For example, the ratio of women taking up studies inthe computing discipline varies per year and region between10% and 35% [3, 21, 38, 44]. Other professions, such aselementary school teaching, nursing, and office administra-tion, are occupied by a higher proportion of women. In fact,it is difficult to find an occupation with a balanced ratioof genders (this also holds for many other social partitions,e.g., ones based on ethnicity or family background). Thisimbalance is the second phenomenon underlying our model.
Homophily. It is a well established social phenomenonthat people tend to associate with others who are similar tothemselves. Characteristics such as gender, ethnicity, age,class background and education influence the relationshipsamong human beings [27] and similarities make communi-cation and relationship formation easier.
In summary, our model is obtained by applying the classi-cal preferential attachment model (see Barabasi and Albert[2]) to a bi-populated minority-majority network augmentedwith homophily. The resulting model is hereafter referred toas the Biased Preferential Attachment Model.
Roadmap. The rest of the paper is organized as follows.In the next section we review related work, then in Sec-tion 3 we introduce the model and the formal definitions ofthe involved properties: glass ceiling, power inequality andhomophily tests. In Section 4 we state our two main theo-rems, and in Section 5 we provide empirical evidence for theexistence of all our necessary ingredients and for the glassceiling effect in a student-mentor network of researchers incomputer science. We conclude with a discussion.
2. RELATED WORKHomophily in social networks. Different characteris-
tics, such as gender, ethnicities, age, class background andeducation, influence the relationships human beings formwith each other [27]. McPherson et al. [32] survey a vari-ety of properties and how they lead to particular patternsin bonding. Gender-based homophily can already be ob-served in play patterns among children at school [31, 41].Eder and Hallinan [12] discovered that young girls are morelikely to resolve intransitivity by deleting friendship choices,while young boys are more likely to add them. Overall,children are significantly more likely to resolve intransitivityby deleting a cross-sex friendship than by adding anothercross-sex friendship [45]. These results show that gender in-fluences the formation of cliques and larger evolving networkstructures. These trends, displaying homophily and genderdifferences in resolving problems in the structure of relation-ships, mean that boys and girls gravitate towards differentsocial circles. As adults, homophilic behavior persists, andmen still tend to have networks that are more homophilicthan women do. This behavior is even more pronounced inareas where they form the majority and in relationships ex-changing advice and based on respect, e.g., mentoring [5, 22,23, 39]. A homophilic network evolution model was studiedin [4]. In this model new nodes connect to other nodes intwo phases. First, they choose their neighbors with a biastowards their own type (the model allows a positive as wellas a negative bias). In a second phase they make an unbi-ased choice of neighbors from among the neighbors of theirbiased neighbors. The authors show that the second phaseovercomes the bias in the first phase and if the second phaseis unbiased, then the network ends up in an integrated state.They illustrate their model with data on citations in physicsjournals.
Gender disparity in science and technology. Gen-der disparities have been observed in the number of profes-sors [13, 34], earnings [40], funding [29] and patenting [10].A related aspect is the “productivity puzzle”: men are moresuccessful when it comes to number of publications andname position in the author list [46], for reasons yet un-clear. Some conjectures raised involve (unknown) biasedperceptions related to pregnancy/child care [6]. E.g., it wasobserved in [34] that science faculty members of both sexesexhibit unconscious biases against women. Gender differ-ences in research output, research impact and collabora-tions was analyzed in a study based on Thomson ReutersWeb of Science databases [26]. It was not only revealedthat papers with women in prominent author positions (soleauthorship, first-authorship and last-authorship) were citedless frequently but the authors also found that age plays animportant role in collaborations, authorship position and ci-tations. Thus many of the trends observed therein might beexplained by the under-representation of women among theelders of science. In other words, fixing the“leaky pipeline”[43]is key for a more equal gender distribution in science.
Minority of women in Computer Science. In thecomputing discipline, the ratio of women taking up stud-ies varies by year and region between 10% and 35% [3,21, 38, 44] ( except in Malaysia, where women form a nar-row majority [35]). This under-representation has been in-vestigated [19, 42, 47] and remedial strategies have beenpropoesd [17, 37]. There is a positive feedback loop [25]:the lack of women leads to a strong male stereotype which
minority & homophily: no minority: no homophily: absolute homophily:r = 0.3, ρ = 0.7 r = 0.5, ρ = 0.7 r = 0.3, ρ = 1 r = 0.3, ρ = 0
(a) (b) (c) (d)
Figure 1: Examples of the Biased Preferential Attachment (BPA) model with various parameter settings.All examples depict a 300-vertex bi-populated network generated by our BPA model starting from a singleedge connecting a blue and a red vertex (with vertex size proportional to its degree).
drives away even more women. It’s been explained that theincrease of the relative number of women in computer sciencewas the best of the investigated remedial strategies, up to a“critical mass” of women. However, as pointed out by Et-zkowitz [14], even achieving a representation of 15% womenmight not guarantee that the effects of a critical mass comeinto play.
3. MODEL AND DEFINITIONS
3.1 Biased preferential attachment modelOur first contribution is in proposing a simple bi-populated
preferential attachment model. In a gist, our model is ob-tained by applying the classical preferential attachment model[2] to a bi-populated minority-majority network augmentedwith homophily. The resulting model is hereafter referredto as the Biased Preferential Attachment Model. Formally,for r ≤ 1/2 and 0 ≤ ρ ≤ 1 let G(n, r, ρ) be a variant of thepreferential attachment model in which the vertices are redor blue, n is the total number of nodes, and r representsthe relative arrival rate of the red vertices (and hence theexpected fraction of red vertices in the network convergesto r as well, as the relative size of the initial population be-comes smaller over time), and ρ represents the level to whichhomophily (incorporated by using rejection sampling) is ex-pressed in the system: for ρ = 1 the system is uniform andexhibits no homophily, whereas for ρ = 0 the system is fullysegregated, and all added edges connect vertex pairs of thesame color.
Let us describe the model in more detail. Denote the so-cial network at time t by Gt = (Vt, Et), where Vt and Et,respectively, are the sets of vertices and edges in the networkat time t, and let δt(v) denote the degree of vertex v at timet (we may omit the parameter t when it is clear from thecontext). The process starts with an arbitrary initial (con-nected) network G0 in which each vertex has an arbitrarycolor, red or blue. (For simplicity we require that a minimalinitial network consists of one blue and one red vertex con-nected by an edge, but this requirement can be removed ifρ > 0). This initial network evolves in time as follows. Inevery time step t a new vertex v enters the network. Thisvertex is red with probability r and blue with probability1 − r. On arrival, the vertex v chooses an existing vertex
u ∈ Vt to attach to according to preferential attachment, i.e.,with probability p proportional to u’s degree at time t, i.e.,P [u is chosen] = δt(u)/
∑w∈Vt
δt(w). Next, if u’s color is the
same as v’s color, then an edge is inserted between v and u;if the colors differ, then the edge is inserted with probabilityρ, and with probability 1 − ρ the selection is rejected, andthe process of choosing a neighbor for v is restarted. Thisprocess is repeated until some edge v, u has been inserted.Thus in each time step, one new vertex and one new edgeare added to the existing graph.
Figure 1 presents four examples of parameter settings forour model on a 300-vertex bi-populated social network. First,Figure 1(a) provides an example for the minority & ho-mophily case with r = 0.3 and ρ = 0.7 so the red verticesare a strict minority in the network and there is some ho-mophily in the edge selection. The next three sub-figurespresent special cases. Figure 1(b) illustrates the no minoritycase (equal-size populations, i.e., r = 0.5) with homophily(ρ = 0.7). Figure 1(c) considers the no homophily case(ρ = 1) with minority (r = 0.3). The last extreme case,shown in Figure 1(d), is absolute homophily, where ρ = 0,but the red vertices are still in the minority (r = 0.3). Thiscase results in fully segregated societies, namely, societieswhere members connect only to members of their own color.In this extreme case, the society in effect splits into two sep-arate networks, one for each of the two populations (exceptfor the single edge connecting the initial red and blue ver-tices).
Consider as an example for our model the social network ofmentor-student relationships in academia. With time, newPhD students arrive, but for some fields female students ar-rive at a lower rate than male students. Upon arrival, eachstudent needs to select exactly one mentor, where the selec-tion process is governed by the mechanisms of preferentialattachment and homophily. Namely, initially the studentselects the mentor according to the rules of preferential at-tachment and then homophily takes its role, rejecting theselection with some probability if their gender is differentand forcing a re-selection. Over time, graduated studentsmay become mentors and some mentors become more suc-cessful than others (in terms of the number of students theyadvise). A glass ceiling effect can be observed in this net-
work if, after a long enough time interval, the fraction offemales among the most successful mentors tends to zero.
We would like to emphasize that the homophily effect thatwe look at is quite minor and “seemingly harmless”, in twoways. First, it is “symmetric”, i.e., it applies both to malestudents with respect to female mentors and to female stu-dents with respect to male mentors. Second, it does notadversely affect the student, in the sense that the studentalways gets admitted in our model. The only tiny (but omi-nous) sign for the potential dangers of this homophilic ef-fect is that it does affect the professor: a male professorwho rejects (or is rejected by) some fraction of the femalecandidates risks little, whereas a female professor who re-jects (or is rejected by) some fraction of the male candidateswill eventually have fewer students overall, since most ofthe applicants are male. In fact, as we show later on, thishomophily-based consequence will only impact her if her fu-ture potential students use preferential attachment to selecttheir mentors.
3.2 Power inequality and glass ceilingOur second contribution is to propose formal definitions
of the glass ceiling effect in social networks. Consider a bi-populated network G(n) consisting of m edges and n nodesof two types, the groups R and B of red and blue nodes.We assume that the network size n tends to infinity withtime. Let n(R) and n(B) denote the number of red and bluenodes, respectively, where n(R) + n(B) = n. The red nodesare assumed to be a minority in the social network, i.e.,denoting the percentage of red nodes in the network by r,we assume 0 ≤ r < 1
2. Let d(R) and d(B) denote the sum
of degrees of the red and blue nodes, respectively, whered(R)+d(B) = 2m. Let topk(R) (respectively, topk(B)) denotethe number of red (resp., blue) nodes that have degree atleast k in G. When G(n) is a random graph, we replacevariables by their expectations in the definitions below, e.g.,we use E[n(R)], E[d(R)], and E[topk(R)]. Next we provideformal definitions for the social phenomena discussed in theintroduction. Power inequality for the minority is defined inthe following way.
Definition 1 (Power inequality). A graph sequenceG(n) exhibits a power inequality effect for the red nodes ifthe average power of a red node is lower than that of a blue(or a random) node, i.e., there exists a constant c < 1 suchthat
limn→∞
1n(R)
∑v∈R δ(v)
1n(B)
∑v∈B δ(v)
=d(R)/n(R)
d(B)/n(B)≤ c . (1)
The definition of the glass ceiling effect is more complex.We interpret the most powerful positions as those held bythe highest degree nodes, and offer two alternative defini-tions. The first tries to capture the informal, “dictionary”definition, which describes a decreasing fraction of womenamong higher degree nodes, i.e., in the tail of the graphdegree sequence. Formally:
Definition 2 (Tail glass ceiling). A graph sequenceG(n) exhibits a tail glass ceiling effect for the red nodes ifthere exists an increasing function k(n) (for short k) suchthat limn→∞ topk(B) =∞ and
limn→∞
topk(R)
topk(B)= 0 .
a
b
c
d
e
f
g
j
h
i
b a c e d j
1
2
3
4
5
6
7
8
f g i h
1
2
3
4
5
6
7
8
(a) (b)
Figure 2: (a) An example bi-populated social net-work with blue and red populations of 6 and 4 ver-tices respectively. (b) The degree sequences of bothpopulations (i.e., the sequence specifying for eachvertex its degree in the network). Considering thetail glass ceiling definition, there are four blue ver-tices of degree greater or equal to 4, but only twosuch red vertices so top4(R)/top4(B) = 1/2. For themoment glass ceiling definition, the second momentfor the blue vertices is 1
6(82+72+52+42+32+32) = 28.6,
while for the red vertices it is 14(72 + 52 + 32 + 32) = 23
and the ratio is 23/28.6. To exhibit a glass ceil-ing, these ratios should converge to zero as the net-work size increases. Regarding homophily, in a ran-dom network with the same population, i.e., 60%blue vertices and 40% red vertices, one expects tofind 36% blue-blue edges, 16% red-red edges and 48%mixed edge. If we take the degree sequences into ac-count we would expect to see 46.8% mixed edge. Inthe above example network we observe only about33% mixed edges, which indicates the effect of ho-mophily.
The second definition considers a more traditional, distribution-oriented measure, the second moment of the two degree se-quences. Formally:
Definition 3 (Moment glass ceiling). A graph se-quence G(n) exhibits a moment glass ceiling g for the rednodes where
g = limn→∞
1n(R)
∑v∈R δ(v)2
1n(B)
∑v∈B δ(v)2
.
When g = 0, we say that G(n) has a strong glass ceiling ef-fect. The intuition behind this definition is that a larger sec-ond moment (and assuming a similar average degree, i.e., nopower inequality) will result in a larger variance and there-fore a significantly larger number of high degree nodes. Aswe show in the full version of the paper, the above two def-initions for the glass ceiling are independent, in the sensethat neither of the effects implies the other.
Testing for homophily in a bi-populated network is basedon checking whether the number of mixed (i.e., red-blue)edges is significantly lower than to be expected if neighborswere to be picked randomly and independently of their color.Formally:
Definition 4 (Homophily Test). [11] A bi-populatedsocial network exhibits homophily if the fraction of mixededges is significantly less than 2r(1− r).
The above definition implicitly assumes that there is powerequality between the colors and therefore is not always ac-curate. A more careful test should take the average degreeof each gender into account.
Definition 5 (Normalized Homophily Test). A bi-populated social network exhibits homophily if the fraction of
mixed edges is significantly less than 2 d(R)2m
(1− d(R)
2m
).
An illustration of these definitions can be found in Figure 2.
4. THEORETICAL RESULTS
4.1 Power inequality and glass ceilingOur main theoretical result (Thm. 4.1) is that in the
biased preferential attachment model, G(n, r, ρ), the glassceiling effect emerges naturally. Additionally, this processgenerates a power inequality, an independent property thatis weaker than the glass ceiling effect. Power inequality de-scribes the situation where the average degree of the minor-ity is lower than that of the majority (although their mem-bers possess the same qualifications). Moreover, we alsoshow (Thm. 4.2) that all three ingredients (unequal entryrate, homophily, preferential attachment) are necessary togenerate what we call a strong glass ceiling effect, i.e., re-moving any one of them will prevent the appearance of aglass ceiling effect. One may suspect that the glass ceilingeffect is in fact a byproduct of power inequality or unequalqualifications; we show in the full paper that this is not thecase. Minorities can have a smaller average degree withoutsuffering from a glass ceiling effect. We also note that ourresults are independent of the starting condition. Even if thenetwork initially consisted entirely of vertices of one color,if a majority of the vertices being added are of the oppositecolor, then eventually the vertices that rise to the highestpositions will be of the new color.
Theorem 4.1. Let 0 < r < 12
and 0 < ρ < 1. ForG(n, r, ρ) produced by the Biased Preferential AttachmentModel the following holds:
1. G(n, r, ρ) exhibits power inequality, and
2. G(n, r, ρ) exhibits both a tail and a strong glass ceilingeffects.
Moreover, all three ingredients are necessary to generatea strong glass ceiling effect.
Theorem 4.2. 1. G(n, r, ρ) will not exhibit a glass ceil-ing effect in the following cases:
(a) If the rate r = 12
(no minority).
(b) If ρ = 1 (no homophily)
(c) If ρ = 0 (no heterophily).
2. G(n, r, ρ) will not exhibit a strong glass ceiling effectif attachment is uniform rather than preferential, i.e.,a new vertex at time t selects an existing vertex toattach to uniformly at random from all vertices presentat time t− 1 (and for any value of r and ρ).
Let us graphically illustrate the above results. Figure 3presents the degree distributions of both the red and blue
populations (as well as of the entire population) for four1,000,000-vertex networks with parameters identical to theexamples in Figure 1. The plots clearly show (and we provethis formally) that in all cases the degree distribution of bothpopulations follows a power-law. (A subset W of verticesin a given network obeys a power-law degree distributionif the fraction P (k) of vertices of degree k in W behavesfor large values of k as P (k) ∼ k−β for parameter β.) Allfigures present (in log-log scale) the cumulative degree dis-tributions, so a power-law corresponds to a straight line (wepresent the samples together with the best-fit line). The-orem 4.1 corresponds to Figure 3(a) with the minority &homophily settings of 0 < r < 1
2and 0 < ρ < 1. In this
case (and only in this case), the power-law exponents of thered and blue populations, β(R) and β(B) respectively, aredifferent, where β(R) > β(B); we prove that this will eventu-ally lead to both tail and strong glass ceiling effect for thered vertices. Theorem 4.2 corresponds to Figures 3(b) and3(c). The figures show that in the case of no minority (i.e.,r = 0.5) or no homophily (i.e., ρ = 1), both β(R) and β(B)are the same (in particular they are equal to 3 as in the clas-sical Preferential Attachment model), and therefore therewill be no glass ceiling effect. Figure 3(d) considers the lastextreme case of absolute homophily. Perhaps surprisingly, inthis case a glass ceiling effect also does not occur, as eachsub-population forms an absolute majority in its own net-work (see again Figure 1(d)). The case of no preferentialattachment (which does not lead to a glass ceiling) is moredelicate and presented in the full version of the paper.
Proof Overview of Theorem 4.1. The basic idea be-hind the proof of Theorem 4.1 is to show that both pop-ulations in G(n, r, ρ) have a power law degree distributionbut with different exponents. Once this is established, itis simple to derive the glass ceiling effect for the popula-tion with a higher exponent in the degree distribution. Tostudy the degree distribution of the red (and similarly theblue) population, we first define αt to be the random vari-able that is equal to the ratio of the total degree of the rednodes (i.e., the sum of degrees of all red nodes) divided bythe total degree (i.e., twice the number of edges). We showthat the expected value of αt converges to a fixed ratio in-dependently of how the network started. The proof of thispart is based on tools from dynamic systems. Basically, weshow that there is only one fixed point for our system. How-ever, determining the expectation of αt is not sufficient foranalyzing the degree distribution, and it is also necessary tobound the rate of convergence and the concentration of αtaround its expectation. We used Doob martingales for thispart. Using the high concentration of the total degree, wewere able to adapt standard techniques to prove the powerlaw degree distribution. Next we give an overview of theproofs and the helping lemmas, but due to space limitationswe defer the details to the full version of this paper.
4.2 Proof sketch of Theorem 4.1 Part 1An urn process. The biased preferential attachment
model G(n, r, ρ) process can also be interpreted as a Polya’surn process, where each edge in the graph corresponds totwo balls, one for each endpoint, and the balls are coloredby the color of the corresponding vertices. When a new (redor blue) ball y arrives, we choose an existing ball c fromthe urn uniformly at random; if c is of the same color as y,then we add to the urn both y and another ball of the same
Both
Female
Male
Β2.9
ΒR3.07ΒB2.86
1 5 10 50 100 500 1000106
105
104
0.001
0.01
0.1
1
Degree
Probabilty
Both
Female
Male
Β3.
ΒR2.99ΒB2.96
1 5 10 50 100 500 1000106
105
104
0.001
0.01
0.1
1
Degree
Probabilty
Both
Female
Male
Β2.91
ΒR2.89ΒB2.9
1 5 10 50 100 500 1000106
105
104
0.001
0.01
0.1
1
Degree
Probabilty
Both
Female
Male
Β2.95
ΒR2.9ΒB2.94
1 5 10 50 100 500 1000106
105
104
0.001
0.01
0.1
1
Degree
Probabilty
minority & homophily: no minority: no homophily: absolute homophily:r = 0.3, ρ = 0.7 r = 0.5, ρ = 0.7 r = 0.3, ρ = 1 r = 0.3, ρ = 0
(a) (b) (c) (d)
Figure 3: Graphical illustrations of our formal claims concerning the glass ceiling effect in the Biased Pref-erential Attachment model. Each figure presents the degree distribution (on a log-log scale) of the red andblue populations from a 1,000,000-vertex network generated by the BPA model with the same parameters asthe corresponding figure in Figure 1. In all cases both populations exhibit a power-law degree distributionbut only in case (a) they with different exponents.
color as c; otherwise (i.e., if c is of a different color), withprobability ρ we still add to the urn both y and another ballof the same color as c, and with probability 1− ρ we rejectthe choice of c and repeat choosing an existing ball c′ fromthe urn uniformly at random. To analyze power inequality,there is no need to keep track of the degrees of individualvertices; the sum of the degrees of all vertices of R is exactlythe number of red balls in the urn.
Denote by dt(R) (respectively, dt(B)) the number of red(resp., blue) balls present in the system at time t ≥ 0.Altogether, the number of balls at time t is dt = dt(R) +dt(B). Initially, the system contains do balls. Noting thatexactly two balls join the system in each time step, we havedt = do + 2t. Note that while dt(R) and dt(B) are randomvariables, dt is not. Recall that balls represent degrees sodt =
∑w∈Vt δt(w).
Denote by αt the random variable equal to dt(R)/dt, thefraction of red balls in the system at time t.
Convergence of expectations. We first claim that theprocess of biased preferential attachment converges to a ratioof α red balls in the system. More formally, we claim thatregardless of the starting condition, there exists a limit
α = limt→∞
E[αt] .
Lemma 4.3. E[αt+1|αt] = αt+F (αt)− αt
t+ 1, where
F (x) =
(1− (1− r) (1− x)
1− x(1− ρ)+ r
x
1− (1− x)(1− ρ)
)/2.
Lemma 4.4. 1. F is monotonically increasing.
2. F has exactly one fixed point, denoted α∗, in [0, 1].
3. The image of the unit interval by F is contained in theunit interval: F ([0, 1]) =
[r2, 1+r
2
]⊂ [0, 1]
4. If x < α∗ then x < F (x) < α∗ and if x > α∗ thenx > F (x) > α∗.
5. α∗ < r.
Now assume αt < α∗. By Lemma 4.4, αt < F (αt) < α∗, soby Lemma 4.3 we obtain αt < E[αt+1|αt] < α∗.
Moreover we can bound the rate of convergence and show:
Lemma 4.5. |α∗ − E[αt]| = O(1/ 3√t).
Theorem 4.6. For any initial configuration, as t goes toinfinity, the expected fraction of red balls in the urn, E(αt),converges to the unique α∗ in [0, 1] satisfying the equation
2α∗ = 1−(1−r) (1− α∗)1− α∗(1− ρ)
+rα∗
1− (1− α∗)(1− ρ). (2)
Hence the limit α∗ is the solution of the cubic equationEq. (3).
(4ρ− 2ρ2 − 2)α3 + (2 + 3ρ2 − 5ρ+ 2r − 2rρ)α2 (3)
+(2ρ− 2r + 2rρ− ρ2)α− rρ = 0
Note that this limit is independent of the initial values doand α0 of the system.
We know that the expected degree of a random vertexis 2 and since the expected degree of a red vertex tends to2α∗/r, which is strictly less than 2 (because of Lemma 4.4Part 5), we can claim:
Corollary 4.7. Let 0 < ρ < 1, 0 < r < 1/2. ThenG(n, r, ρ) has a power inequality effect.
4.3 Proof sketch of Theorem 4.1 Part 2Concentration. To prove the glass ceiling effect we first
bound the degree distribution. To do this we need to boundthe rate by which dt(R) converge to α · t. Let Xi ∈ 0, 1, 2be the number of new red balls in the system at time i. Notethat dt(R) =
∑t0Xi. Let
Ψi = EXi+1,Xi+2,...,Xt
[ t∑j=0
Xj |X1, X2, . . . , Xi].
Observe that (Ψi)i is a Doob Martingale [33], and note that
Ψ0 = E[∑t
i=0Xi]
= E[dt(R)
].
Theorem 4.8 (Azuma’s inequality [1]). Let Ψt be amartingale such that for all i, almost surely |Ψi−Ψi−1| < ci.Then for all positive t and all positive reals x,
Pr(Ψt −Ψ0 ≥ x) ≤ exp
(−x2
2∑i c
2i
).
Lemma 4.9. Let Ci = |Ψi −Ψi−1|. Then Ci = O(√t/i).
For simplicity of the description, let us assume hereafterthat do = 0, hence αt = dt(R)/(2t). By Theorem 4.8 andLemma 4.9 we have
Lemma 4.10. Pr[|dt(R)−2tE(αt)| > O(2
√t log t)
]≤ 1
t4.
Combining Lemmas 4.5 and 4.10 yields:
Corollary 4.11.
Pr
[|α∗ − αt| > max
2 log t√
t,
13√t
]<
1
t4.
Degree distribution. We investigate the degree distri-bution of the red and blue vertices in a graph generated bythe above described process, following the analysis outlineof [7] for the basic preferential attachment model.
Let mk,t(B) (resp., mk,t(R)) denote the number of blue(resp., red) vertices of degree k at time t. For x ∈ R, B,define
Mk(x) = limt→∞
E(mk,t(x))
t. (4)
Theorem 4.12. The expected degree distributions of theblue and red vertices follow a power law, namely, Mk(B) ∝k−β(B) and Mk(R) ∝ k−β(R). If 0 < r < 1/2 and 0 < ρ < 1then β(R) > 3 > β(B).
Equipped with Theorem 4.12, Part 2 of Theorem 4.1 fol-lows easily. Indeed, for the tail glass ceiling effect, let k(n) =
n1β(R) . Then
E[topk(R)] = n(R)∑k′≥k
Mk′(R) ,
E[topk(B)] = n(B)∑k′≥k
Mk′(B).
For k′ = n1β(R) we have nMk′(R) = O(n · n−
β(R)β(R) ) = O(1)
while nMk′(B) = Ω
(n · n−
β(B)β(R)
)= Ω(n
1− β(B)β(R) ) = Ω(nε)
for ε > 0. The result then follows since n(R) < n(B) andMk′(R) < Mk′(B) for k′ > k.
For the moment glass ceiling effect we can show similarly:
g = limn→∞
∑k2Mk(R)∑k2Mk(B)
= limn→∞
O(n3−β(R))
Ω(n3−β(B))
= limn→∞
O
(1
nε′
)= 0
for some ε′ > 0.The rest of this section sketches a proof of Theorem 4.12.
Note that m0,0(B) = do(B). We derive a recurrence forE(mk,t(B)). A blue vertex of degree k at time t could havearisen from three scenarios: (s1) at time t−1 it was alreadya blue vertex of degree k and no edge was added to it attime t. (s2) at time t−1 it was a blue vertex of degree k−1and an edge was added to it at time t. (s3) in the specialcase where k = 1, at time t − 1 it did not exist yet and ithas arrived as a new blue vertex at time t. Thus letting Ftbe the history of the process up to time t, for any k > 1, theexpectation of mk,t+1(B) conditioned on Ft satisfies
E(mk,t+1(B)|Ft) =
mk,t(B)
(1−
rdt(B)ρk
dt(B)
dt(R)+dt(B)ρ−
(1−r)dt(B) kdt(B)
dt(R)ρ+dt(B)
)+ mk−1,t(B)
(rdt(B)ρ
k−1dt(B)
dt(R)+dt(B)ρ+
(1−r)dt(B) k−1dt(B)
dt(R)ρ+dt(B)
).
For k = 1 we similarly have
E(m1,t+1(B)|Ft) =
m1,t(B)(
1− ρrdt(R)+dt(B)ρ
− 1−rdt(R)ρ+dt(B)
)+ (1− r) .
Recalling again that αt = dt(R)/(2t), the above can berewritten as
E(mk,t+1(B)|Ft) =
mk,t(B)(
1− rρk2t(αt+(1−αt)ρ) −
(1−r)k2t(αtρ+(1−αt))
)+ mk−1,t(B)
(rρ(k−1)
2t(αt+(1−αt)ρ) + (1−r)(k−1)2t(αtρ+(1−αt))
)and for k = 1,
E(m1,t+1(B)|Ft) =
m1,t(B)(
1− ρr2t(αt+(1−αt)ρ) −
1−r2t(αtρ+(1−αt))
)+ (1− r).
This can be expressed as
E(mk,t+1(B)|Ft) = mk,t(B)
(1−At
k
t
)(5)
+mk−1,t(B)Atk − 1
t,
E(m1,t+1(B)|Ft) = m1,t(B)
(1− At
t
)+ (1− r), (6)
using the notation
At =rρ
2αt + 2(1− αt)ρ+
(1− r)2αtρ+ 2(1− αt)
.
Note that At is a random variable so we next bound itsdivergence. Let
CB =rρ
2α+ 2(1− α)ρ+
(1− r)2αρ+ 2(1− α)
CR =(1− r)ρ
2(αρ+ 1− α)+
r
2(α+ (1− α)ρ)).
We have
Lemma 4.13. Pr
[|At − CB| > max
2 log t√
t,
13√t
]<
1
t4.
A similar claim can be made for CR. This enables us toestablish the following.
Lemma 4.14.
• M1(B) exists and equals (1− r)/(1 + CB),
• For k ≥ 2, Mk(B) exists and equalsMk−1(B) · (k − 1)CB/(1 + kCB),
• M1(R) exists and equals r/(1 + CR), and
• For k ≥ 2, Mk(R) exists and equalsMk−1(R) · (k − 1)CR/(1 + kCR),
It is possible to show the following about CB and CR:
Lemma 4.15.
• If 0 < r < 1/2 and 0 < ρ < 1 then CR <12< CB
• If r = 1/2 then CR = CB = 1/2.
• If ρ = 0 or ρ = 1 then CR = CB = 1/2.
To show that the degree distributions of both the red andthe blue vertices follow power laws we recall that a powerlaw distribution has the following property: Mk ∝ k−β forlarge k, where β is independent of k. If Mk ∝ k−β , then
Mk
Mk−1=
k−β
(k − 1)−β=
(1− 1
k
)β= 1− β
k+O
(1
k2
).
Solving for the blue vertices, Mk(B) and the blue exponentβ(B), and using Lemma 4.14, we get:
Mk(B)
Mk−1(B)=
(k − 1) · CB
1 + k · CB
= 1− CB + 1
k · CB + 1
= 1−1 + 1
CB
k+O
(1
k2
)hence β(B) = 1 + 1/CB. Similarly, for red vertices of degreek, Mk(R) decays according to a power law with exponentβ(R) = 1 + 1/CR. Note that when CR <
12< CB we have
β(R) > 3 > β(B) thus proving Theorem 4.12.
5. EMPIRICAL OBSERVATIONSTo provide empirical evidence illustrating the results of
our analysis in real-life, we studied a mentor-student networkof researchers in computer science, extracted from DBLP[28], a dataset recording most of the publications in com-puter science. A filtering process, described in detail in thefull version of this paper, creates a list of edges connectingstudents to mentors. For each edge we determined the gen-der of the student and the mentor and the year in which theconnection was established. The resulting network spansover 30 years and has 434232 authors and 389296 edges. Asmay be expected based on previously reported studies, ourmentor-student network exhibits a minority-majority parti-tion (namely, a low proportion of 21% females), homophily,power law distribution and a glass ceiling effect.
Figure 4(a) reveals that over time, the fraction of femalesin the network (n(R)/n, the shaded red area) has increased,but it is still below 21%. Also the average degree for femalesvertices is lower (1.48 vs 1.87). Figure 4(b) presents an in-dication for homophily in the mentoring selection process.This is done by the homophily test of [11], which comparesthe expected number of “mixed” (female-male) edges to theobserved one (see also Section 3.2).
Figure 5 presents indications for the glass ceiling effect.Figure 5(a) shows that the fraction of females among thevertices of degree k or higher, namely, topk(R)/topk(B), de-creases continuously as k increases. The first major decreaseoccurs when moving from the group of “students” (i.e., de-gree 1 vertices) to the group of researchers of degree 2 orhigher: the fraction of females drops from top1(R) ≈ 21%to top2(R) < 15%. It is important to note that the dataindicates that even at the high end of the graph, a few fe-male researchers with very high degrees are still present;however, our definitions for the glass ceiling ignore this ex-tremal effect, which is caused by a few individuals, and con-centrate on the averages over large samples. Indeed, whenthe sample size is large enough, the fraction of the femaleresearchers decreases. Figure 5(b) shows a strong indica-tion that the degree distribution of the vertices (females,males and combined) follows a power law. This in turn isassociated with a preferential attachment mechanism thatis known to result in a power law degree distribution. Notethat the power-law exponent β for the graph of the female
1980 1985 1990 1995 2000 2005 2010Year
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20
40
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80
100
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observed M power %
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(a)
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of
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dges
E[mixed edges] observed female fraction,assume equal degree and no homophily
E[mixed edges] observed degree, assume no homophily
observed mixed edges
(b)
Figure 4: Female power and homophily in the com-puter science mentor graph. (a) The rate (i.e., per-centage in the population) of females over time,compared with their normalized power, defined asd(R)/(2m). Males have more power than expected bytheir rate, while females have less power than ex-pected by their rate. (b) Evidence for homophily:a comparison of the observed number of “mixed”edges to the expected value assuming there is nohomophily. We consider two cases: (i) the ex-pected number of mixed edges ignoring the differ-ence between the male and female average degree(expected: 127963.09 std: 293.08) and (ii) the ex-pected number of mixed edges while considering thedifferent degree sequences for males and females (ex-pected: 110777.11 std: 281.52). In both cases theobserved value (101607 edges) significantly deviatesfrom the expectation (the error bars indicate theexpected value ± 10 times the standard deviation)with extremely low p-values.
researchers is β = 2.91 (in the best fit), which is higherthan the corresponding exponent in the graph for the maleresearchers, β = 2.58. Our analysis (presented in 4.2 and4.3) establishes that if the degree distribution of both sub-populations follow a power law and the exponent for theminority sub-population is higher than that of the majoritysub-population, then a strong moment glass ceiling effectwill appear.
6. DISCUSSIONOne obvious limitation of our model is that it is some-
what simplistic and captures only one possible mechanismfor generating a glass ceiling effect. It ignores many impor-
1 2 5 10 20 50 100Degree (Number of Mentees + 1)
0
5
10
15
20
25
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enta
ge o
f Fem
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Rese
arc
hers
am
ong R
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arc
hers
wit
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egre
e a
t le
ast
x
(a)
1 2 5 10 20 50Degree
10−5
10−4
10−3
10−2
10−1
1
Pro
babi
lity
BothMaleFemale-2.62-2.58-2.91
(b)
Figure 5: Glass ceiling effect in mentor graph: (a)percentage of females in the mentor population ofdegree at least k. Female start with 21% in thepopulation and drop to below 15% when consideringdegree at least 2 (faculty members). It continuesto decrease (ignoring small samples at the end, seetext). Vertex size and darker color represent largersample space. (b) The power-law-like degree distri-bution for both females and males. The exponent βfor females is higher than for males, demonstratingthe glass ceiling effect.
tant aspects of real life (such as sexual tension, fear, familyresponsibilities and jealousy, to name a few) and alterna-tive (co-existing) mechanisms that contribute to the effect.For instance, our model cannot be used to explain the oc-currence of a glass ceiling effect in contexts where pairwiseindividual interactions play a less dominant role than inacademia. To account for the glass ceiling effect in suchcontexts as well as others, one may consider alternative ex-planations. In particular, a common possible explanation isthe “leaky pipeline” phenomenon, namely, the phenomenonthat women tend to quit or slow down their careers in orderto invest more time in their families. This phenomenon canbe modeled mathematically in several different ways. Onesuch way is by introducing vertex departures in addition tovertex arrivals, with a bias in the form of increased depar-ture rate of the minority group. But in fact, such a dynamic“leaky pipeline” model allows several reasonable sub-modelsthat will not generate a glass ceiling effect, as well as someother sub-models that do. Moreover, the cause and effect re-lationships between glass ceiling and leaky pipeline are notnecessarily one-directional; while the glass ceiling effect mayindeed be the outcome of the “leaky pipeline” phenomenon
in certain settings, there are other settings where it may beits (partial) cause. An interesting direction for future workwould be to describe a more complete model, most likelycombining a number of different mechanisms contributingjointly to the glass ceiling effect. In any case, we find itremarkable that the simple mathematical mechanism pre-sented here (based on homophily) is sufficient to explain (atleast parts of) the glass ceiling effect, despite the fact thatit does not utilize the “leaky pipeline”.
Our Findings may suggest ways to deal with the glass ceil-ing phenomenon. By better understanding the roots of theglass ceiling effect, one can address each of the elements andattempt to mitigate them or deal with those elements thatare easier to manage. Our research indicates that for cer-tain mechanisms involved in the formation of a glass ceiling,removing one element may eliminate the glass ceiling effect.Hence, while it might be difficult to modify the human ten-dencies of homophily and preferential attachment one couldattempt to balance the proportions of minorities within thepopulation or impose a proportional representation of suc-cessful women at the top level. Both of these options maybe classified as variants of affirmative action, but the latter,even if more common, seems to avoid the roots of the prob-lem. In particular, a more equally represented society couldbe created by encouraging minorities to enter the system,as our findings indicate that increasing the ratio of minori-ties at the entry stage may mitigate the glass ceiling effectat least partially. This conclusion is in line with a commonview [36, 43], which states that fixing the “leaky pipeline”is key for a more equal gender distribution in science. Bydetermining and examining the causes of the glass ceilingeffect, we can work on alleviating the glass ceiling effect,resulting in a richer and more diverse community.
AcknowledgementsThe authors thank Eli Upfal for suggesting the use of a DoobMartingale and the anonymous reviewers of this paper fortheir helpful comments.
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