What Is The Matching Hypothesis

PLoS Ane. 2015; ten(vi): e0129804.

An Analysis of the Matching Hypothesis in Networks

Tao Jia,^{i ,} ^{ii ,} ^{3 ,} ^* Robert F. Spivey,^{3 ,} ⁴ Boleslaw Szymanski,^{ane ,} ^{2 ,} ^v and Gyorgy Korniss^{1 ,} ³

Tao Jia

^one Social Cerebral Networks Academic Inquiry Centre, Rensselaer Polytechnic Institute, Troy, NY, 12180 USA

² Section of Computer Science, Rensselaer Polytechnic Plant, Troy, NY, 12180 Us

^three Section of Physics, Practical Physics and Astronomy, Rensselaer Polytechnic Constitute, Troy, NY, 12180 U.s.

Robert F. Spivey

^three Section of Physics, Practical Physics and Astronomy, Rensselaer Polytechnic Establish, Troy, NY, 12180 USA

^iv Department of Electric and Computer Engineering, Duke Academy, Durham, NC, 27708 U.s.

Boleslaw Szymanski

^one Social Cerebral Networks Academic Inquiry Heart, Rensselaer Polytechnic Institute, Troy, NY, 12180 USA

² Department of Reckoner Science, Rensselaer Polytechnic Plant, Troy, NY, 12180 USA

^five Społeczna Akademia Nauk, Łódź, Poland

Gyorgy Korniss

¹ Social Cognitive Networks Academic Enquiry Center, Rensselaer Polytechnic Plant, Troy, NY, 12180 USA

³ Section of Physics, Practical Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, NY, 12180 Usa

Lidia Adriana Braunstein, Academic Editor

Received 2015 Mar xvi; Accustomed 2015 May 13.

Abstract

The matching hypothesis in social psychology claims that people are more probable to form a committed relationship with someone equally attractive. Previous works on stochastic models of human mate selection procedure indicate that patterns supporting the matching hypothesis could occur even when similarity is not the primary consideration in seeking partners. Even so, most if not all of these works concentrate on fully-continued systems. Here we extend the analysis to networks. Our results betoken that the correlation of the couple's attractiveness grows monotonically with the increased average caste and decreased degree diverseness of the network. This correlation is lower in sparse networks than in fully-connected systems, because in the one-time less attractive individuals who find partners are likely to be coupled with ones who are more attractive than them. The hazard of failing to be matched decreases exponentially with both the attractiveness and the degree. The matching hypothesis may not hold when the degree-attractiveness correlation is present, which can give rise to negative attractiveness correlation. Finally, we detect that the ratio between the number of matched couples and the size of the maximum matching varies non-monotonically with the average degree of the network. Our results reveal the function of network topology in the process of man mate choice and bring insights into time to come investigations of different matching processes in networks.

Introduction

The procedure of pairing and matching between members of two disjoint groups is ubiquitous in our society. The underlying mechanism can be purely random, simply in full general decisions on selections are guided past rational choices, such as the relationship between advisor and advisee, the employment between employer and employee and the marriage between heterosexual male and female individuals. In many of these cases, similarities between the two paired parties are widely observed, such as similar research interests betwixt the counselor and advisee and matched marketplace competitiveness between the executives and the company. The principle of homophily, the tendency of individuals to associate and bond with others who are like to them, tin be practical to explain such similarities [1]. Still, in some cases dissimilar mechanisms may be at work in addition to simply seeking similarities. For case, information technology has been discovered that people end upwardly in committed relationship in which partners are likely to be of similar attractiveness, equally predicted by the matching hypothesis in the field of social psychology [ii, 3]. Yet, if the closeness in attractiveness is the goal when searching for partners, i needs an objective self-interpretation of it, which is rarely the instance [iv]. Furthermore, it is found in social experiments that people tend to pursue or accept highly desirable individuals regardless of their own attractiveness [three, 4]. These findings suggest that the observed similarities may non exist solely acquired past explicitly seeking similarities. In some previous works, stochastic models are applied to simulate the process of human being mate pick [5–x]. By simply assuming that highly attractive individuals are more than likely to be accepted, the arrangement generates patterns supporting the matching hypothesis even when similarity is not directly considered in the partner choice process [5]. Yet, most if not all of these works (with a few recent exceptions [11–13]) concentrate on systems without topology, also known as fully-connected systems, in which one connects to all others in the other party and competes with all others in the aforementioned party. In reality, however, one knows only a limited number of others as characterized by the degree distribution of the social network. Hence a simple merely fundamental question arises: what is the upshot of the matching process when topology is present?

In this piece of work, we aim to address this question past analyzing the touch of network construction on the specific example of the process of matching, namely, homo mate choice. Our motivation to address this question is caused non merely by the limited cognition on this matter, just too by the fact that topology could fundamentally change properties of the organisation and farther touch on its dynamical process. We take witnessed evidence of such impact, accumulated in the last decades from the advances towards agreement complex networks: a few shortcuts on a regular lattice can drastically reduce the mean separation between nodes and give rise to the small-globe phenomenon [14, xv], the power-law degree distribution of scale-free networks can eliminate the epidemic threshold of epidemic spreading [xvi, 17] and synchronization can be reached faster in networks than in regular lattices [18–xx]. Indeed, numerous discoveries have been fabricated in different areas when because topology in the analysis of many classical problems [21–30]. Hence it is fair to expect that the network topology would also bring new insights on the matching process that we are interested in.

Methods

We start with a bipartite graph with 2N nodes. The bipartite graph consists of 2 disjoint sets m and f of equal size, representing two parties, each with N members. While our model can be more general, for simplicity, we consider the two parties as collections of heterosexual male and female individuals (Fig 1a). Each node, representing 1 individual, has grand links drawn from the degree distribution P(k), randomly connecting to thou nodes in the other set up. On average, a node has ⟨grand⟩ = ∑kP(g) links, referred to the boilerplate degree of the network. To characterize the process of homo mate choice, each node is assigned a random number a equally its bewitchery drawn uniformly from the range [0,one). Combining features in some previous works [5, 8] with the network construction, we consider the process of man mate choice as a two-pace stochastic process which generates the numerical model as follows (Fig 1b):

At each detached time pace, randomly pick a link. Let'due south denote the nodes continued by this link every bit node i and node j and their attractiveness as a _i and a _j, respectively.
Draw two random numbers independently and uniformly from the range [0,1), denoted by r _i and r _j. Check the matching condition defined as a _i > r _j and a _j > r _i.
If the matching condition is satisfied and nodes i and j are not in a relationship with each other, pair them into intermediate pairing and dissolve them from whatsoever previous intermediate pairing with other nodes, if there are whatsoever.
If the matching status is satisfied and nodes i and j are already in the intermediate pairing with each other, join them into the stable couple. Make nodes i and j unavailable to others by removing them from the network together with all their links.
Repeat from step 1 until at that place is no link left.

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(a) An example of a bipartite graph, which is composed of two disjoint sets of nodes m and f. There is no link betwixt nodes in the aforementioned set and the connectedness betwixt sets is characterized by degree distribution P(g). (b) The activeness scheme of the mate choosing process. 2 nodes i and j accept to undergo an intermediate stage to attain the stable long term relation. During the intermediate stage nodes i and j are also bachelor to build relationship with other nodes. If this happens they intermission and their relationship is dorsum to the initial land.

The matching condition in footstep two ensures that individuals mutually accept each other. The decision making is probabilistic: the probability that node i accepts node j is a _j (independent of its own attractiveness a _i). A pairing is successfully established only when both individuals decide to take each other. The intermediate pairing created in footstep three corresponds to the trend of people not to fully commit to a relationship at the beginning and to form a stable couple only after such unstable intermediate stage. The removal of nodes and links in stride 4 merely accelerates the simulation, as these links should not be considered by others and the respective nodes in the stable state are not available for matching. Undoubtedly our model only captures a very pocket-sized fraction of features in the matching process. The goal of this piece of work is non to suggest a sophisticated model that is able to regenerate all observations in reality. Instead, we focus on attractiveness and popularity (degree) that are essential in this process, hence this model could be the simplest to study the interplay between these two factors, shedding light on the effect of topology on this process.

To report the effects of topology, we focus on iii most commonly used network structures with dissimilar caste distributions. 1) random k-regular graph (RRG) whose degree distribution follows a delta function P(k) = δ(k−⟨k⟩), where ⟨m⟩ is the average degree of the network, corresponding to an farthermost example that each person knows exactly the same number of others; ii) Erdős-Rényi network (ER) with a Poisson caste distribution P(k) = e ^{−⟨thousand⟩}⟨m⟩^k/k!, representing the state of affairs that most nodes have similar number of neighbors and nodes with very high or low degrees are rare [31]; 3) scale-free network (SF) generated via static model whose degree distribution has a fat-tail P(k) ∼ thou ^−γ, featuring a big number of low degree nodes and few high caste hubs [32, 33]. The constructions of these networks are equally follows.

Constructing a random k-regular graph. We start from two sets (sets m and f) of N disconnected nodes indexed past integer number i (i = i,…N). For each node i in the set thousand, connect it to nodes i, i+i, … and i+k−1 in the set f (using periodic boundary condition such that node N in the set m connects to node North, 1, … and yard−two in the prepare f, and and so on). Then randomly pick two links, assuming that 1 link connects nodes i in the ready m and j in the prepare f and the other connects nodes i′ in the set up m and j′ in the set f. Check if there is a connection betwixt nodes i and j′ and nodes i′ and j. If not, remove original links and connect nodes i and j′ and nodes i′ and j. Repeat this process sufficiently big number of times such that connections of the network are randomized.

Amalgam an Erdős-Rényi network. We showtime from two sets (sets one thousand and f) of N disconnected nodes indexed by integer number i (i = i,…Due north). Randomly select two nodes i and j respectively from sets thou and f. Connect nodes i and j if there is no connection between them. Repeat the process until Due north⟨chiliad⟩ links are created.

Amalgam a scale free network. The scale-free networks analyzed are generated via the static model. Nosotros start from 2 sets (sets chiliad and f) of Due north asunder nodes indexed past integer number i (i = i,…N). The weight w _i = i ^−α is assigned to each node, where α is a real number in the range [0,1). Randomly selected two nodes i and j respectively from sets k and f, with probability proportional to w _i and west _j. Connect nodes i and j if there is no connectedness between them. Repeat the process until N⟨k⟩ links are created. The degree distribution under this structure is $P (k) = \frac{{[⟨ grand ⟩ (1 - α) / 2]}^{1 / α}}{α} \frac{Γ (k - 1 / α, ⟨ k ⟩ (i - α) / 2)}{Γ (k + 1)}$ where Γ(south) the gamma function and Γ(south, x) the upper incomplete gamma function. In the large k limit, the distribution becomes $P (1000) \sim k^{- (1 + \frac{1}{α})} = k^{- γ}$ .

Introducing correlations between the attractiveness and the degree. We generate iiN random numbers drawn between 0 and ane and sort them in ascending order and index them by integer number i (i = ane, … 2N). We sort nodes of networks in ascending order of their degrees and index them by integer number j (j = 1, … twoNorthward). For positive correlation between the degree and attractiveness, assign i ^th random number as the attractiveness of node j = i. For negative correlation betwixt the degree and attractiveness, assign i ^th random number as the bewitchery of node j = twoN−i+1.

Results

Effects of Network Topology on the Correlation in Bewitchery

The matching hypothesis suggests similarities in attractiveness between the two coupled individuals. To test it, nosotros utilise the Pearson coefficient of correlation ρ every bit a measure of similarity, that is defined every bit

$\begin{matrix} ρ = \frac{\sum_{i}^{n} (a_{m, i} - {\bar{a}}_{m}) (a_{f, i} - {\bar{a}}_{f})}{\sqrt{\sum_{i}^{n} {(a_{1000, i} - {\bar{a}}_{m})}^{2}} \sqrt{\sum_{i}^{n} {(a_{f, i} - {\bar{a}}_{f})}^{2}}}, \end{matrix}$

(1)

where a _{m, i} and a _{f, i} are the attractiveness of the individuals in sets yard and f of the i ^th couple, ${\bar{a}}_{one thousand}$ and ${\bar{a}}_{f}$ are the boilerplate attractiveness of the matched individuals in sets m and f and n is the number of matched couples in the network. The Pearson coefficient of correlation ρ varies from -1 to 1, where 1 corresponds to the strongest positive correlation when two quantities are perfectly linearly increasing with each other, whereas -ane is the strongest negative correlation when ii quantities are perfectly linearly dependent and 1 decreases when the other increases.

We first check the scenario studied in near of the previous works, when topology is not considered and each node is potentially able to lucifer an capricious node in the other set. Our model generates a high correlation of the couple's attractiveness with the boilerplate ρ ≈ 0.56 (Fig 2a). This value is similar to the effect generated in the previously proposed model which accounts also for bewitchery decay [5] fifty-fifty though this feature is not present in ours. It is noteworthy that similarity is non explicitly considered when establishing a matching in this model and each individual only seeks attractive partners. Withal, the mutual agreement between two individuals finer depends on the articulation attractiveness of both. Hence individuals with loftier bewitchery will have the advantage in finding highly attractive partners, causing them to be removed from the dynamics soon, while less attractive individuals observe their matches later. Therefore, as time goes on, only less and less attractive individuals are available to form a couple, thus they are more likely to get a partner with similar attractiveness.

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(a) The Pearson coefficient of correlation ρ of the bewitchery between the two coupled individuals in different systems. ρ is strongest in fully-connected systems. In sparse networks, ρ increases monotonically with the average degree ⟨k⟩ and decreases with the caste multifariousness. For all cases investigated, system size is iiDue north and Due north = x,000. (b) The average attractiveness ${\bar{a}}_{f}$ of individuals in the set up f who are matched with those in a subset of k with attractiveness in the range [a _thou−0.05, a _g+0.05) for a series of points a _m. In fully-connect systems, the less attractive individuals are bound to exist coupled with ones who are also less attractive. In sparse networks, however, they are coupled with ones who are more than attractive. (c) The attractiveness contour figure of the coupled individuals in Erdős-Rényi networks with average degree ⟨thou⟩ = 5. A pattern emerges even when similarity is not the motivation in seeking partners. a _m and a _f are the bewitchery of nodes in sets m and f, respectively. (d) The attractiveness contour figure of the coupled individuals in fully-connected systems. The correlation is strongest towards the less attractive individuals (the circled office).

The positive correlations in attractiveness are as well observed in all three classes of networks studied. They are lower than the correlation observed in the fully-continued systems just increase monotonically with the boilerplate caste ⟨g⟩. Furthermore, as the network degree distribution varies from a delta role to a Poisson distribution and to a fat-tail distribution, the variance in the degree distribution increases. Our results indicated that for a given ⟨k⟩, ρ decreases with the increased degree diversity (Fig 2a). In other words, the broader the degree distribution is, the lower the correlation in attractiveness between the two coupled individuals will be. The reason is that as the degree diversity increases, more and more links are connected to a few high caste nodes. The bulk of nodes accept lower degrees compared to the network with the same degree but smaller degree diversity. Hence the bulk of nodes take less opportunities in selecting partners and therefore smaller chance to find a partner with closely matched bewitchery. As the result the bewitchery correlation decreases.

While the correlation in attractiveness is strongest when the organization is fully-connected, we notice that the departure in the correlations is caused more often than not by the matched individuals with depression bewitchery. Indeed, the average attractiveness of those who are coupled with highly desired individuals does not depend much on the presence of the network construction (Fig 2b–2d). In fully-connected systems, less attractive individuals are bound to be coupled with partners of low attractiveness, which contributes significantly to the total correlation ρ. In sparse networks, however, if they successfully discover partners, their partners are probable to be more attractive than them. Therefore, the limited pick in sparse networks reduces competitions among individuals, especially for those with low bewitchery, hence giving rise to lower attractiveness correlations betwixt the two coupled individuals.

In fully-connected systems all individuals are able to discover their partners. Only in networks one faces a chance of failing to be matched. How ofttimes it occurs depends on 1'southward popularity (degree) and attractiveness. Here we consider P _non(a, k) defined every bit the probability of failing to be matched conditioned on degree k and attractiveness inside the range [a−0.05, a+0.05). We discover that P _non(a, g) drops exponentially with both degree k and bewitchery a. This implies that getting more popular brings the similar benefit as beingness more bonny in terms of finding a partner (Fig 3).

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(a, b) The probability of declining to be matched conditioned on attractiveness a and caste g (P _not(a, k)) decreases exponentially with a and k in scale-free networks with P(k) ∼ k ^−γ, γ = 3 and ⟨k⟩ = 5.

So far we have concentrated only on cases where at that place is no correlation betwixt one'due south popularity (degree) and attractiveness. In reality these two features are frequently correlated. On ane hand, the positive correlation is somewhat expected as a highly bonny person can potentially be also very popular hence having a larger caste. On the other paw, negative correlation could also occur when those with depression attractiveness are more agile in making friends to residuum their disadvantage in attractiveness. We extend our analysis to two extreme cases when degree and attractiveness are correlated (see Method). For a given network topology, the correlation of attractiveness (ρ) is strongest when the degree and the attractiveness are positively correlated and weakest when they are negatively correlated. Information technology is noteworthy that with negative caste-bewitchery correlation, ρ can go negative in networks with low ⟨k⟩, suggesting that the matching hypothesis may non concur in such networks even though the underlying mechanism does not change (Fig four).

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The Pearson coefficient of correlation ρ of the attractiveness between the 2 coupled individuals in Erdős-Rényi networks with size 2N (N = 10,000) and varying boilerplate caste ⟨k⟩.

ρ increases monotonically in all three cases analyzed. Notwithstanding, ρ is largest in networks where the degree and the attractiveness are positively correlated. When they are negatively correlated, ρ is weakest and can even be negative.

Number of Couples Matched

Another quantity affected by topology and typically studied is the number of couples a system can somewhen lucifer n[xiii, 34]. When the system is fully-connected, everyone can observe a partner and the number of couples is n = N. In thin networks, typically in that location are fewer matched couples than N and the highest number of matched couples n _max is given by the maximum matching which disregards the attractiveness [35, 36]. To measure the performance of the system in terms of the matching, we focus on the quantity R = due north/n _max defined as the ratio between the number of couples matched and the size of the maximum matching. While both the number of the couples matched and the size of the maximum matching increase monotonically equally the network becomes denser (Figs 5a, 5b), their ratio R changes non-monotonically with ⟨k⟩ (Fig 5c). The organisation's functioning can be relatively adept when the network is very thin or very dense, but relatively poor for the intermediate range of density. This is mainly because when more links are added to the system, the number of couples matched increases slower than the size of the maximum matching; only when this size becomes saturated to N the ratio R starts to increase with ⟨thou⟩.

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(a) The size of the maximum matching n _max increases monotonically with the average degree ⟨k⟩ in different networks. (b) The number of matched couples due north increases monotonically with the average degree ⟨k⟩ in different networks. (c) The ratio betwixt the number of matched couples and the size of the maximum matching (R = due north/n _max) varies not-monotonically with the average caste ⟨grand⟩. (d) Different behaviors of R in Erdős-Rényi networks where the correlation between caste and the attractiveness varies. Negative correlation between the caste and the attractiveness yields the largest R while positive correlation between the caste and the attractiveness results in the smallest R. Networks tested in all cases are with size 2Due north (N = 10,000).

Correlation between the degree and attractiveness too plays a role in the value of R achieved by a network. The maximum matching n _max depends only on the topology of the network and does not depend on the attractiveness. A successful matching betwixt two nodes in our model, however, depends on both their attractiveness and their degrees. Therefore, R depends on the degree-attractiveness correlation. In both cases when either positive or negative correlation between degree and bewitchery is present, R varies non-monotonically with ⟨k⟩ just like in the case when there is no degree-bewitchery correlation (Fig 5d). However, negative correlation between degree and attractiveness yields more while positive correlation yields fewer matched couples than that when degree and attractiveness are uncorrelated. Considering the fact that the similarity between the 2 coupled individuals (ρ) is largest in networks with positive degree-attractiveness correlation and smallest with negative caste-attractiveness correlation, such a dependence of R on caste-attractiveness correlation implies that the system's performance in terms of the number of matched couples is better when it is less selective.

Discussion

In summary, we studied the effect of topology on the process of human mate pick. In full general, our findings support the conclusion of the previous works that similarities in attractiveness between coupled individuals occur fifty-fifty though the similarity is non the master consideration in searching for partners and each individual merely seeks attractive partners, in agreement with the matching hypothesis. When topology is nowadays, the extent of such similarity, measured by Pearson coefficient of correlation, grows monotonically with the increased boilerplate degree and decreased degree diverseness of the network. The correlation is weaker in sparse networks because in them the less attractive individuals who are successful in finding partners, are likely to be coupled with more attractive mates. In fully-connected systems, still, they are nearly certain to be coupled with partners also less attractive, contributing significantly to the total attractiveness correlation.

Another issue of the topology is that ane faces a chance of failing to notice a partner. Such the run a risk decays exponentially with i's attractiveness and degree, therefore being more popular can bring benefits in terms of finding a partner like to beingness more attractive. The correlation of couple'south attractiveness is also affected by the degree-bewitchery correlation, which is strongest in networks where attractiveness and popularity are positively correlated and weakest when they are negatively correlated. In networks with negative caste-bewitchery correlation, the attractiveness correlation betwixt coupled individuals can be negative when the average degree is depression, implying that matching hypothesis may non hold in such systems. Finally, the number of couples matched too depends on the topology. The ratio between the number of matched couples and the maximum number of couples that can be matched, denoted every bit R, changes not-monotonically with the average caste. R is largest in networks with negative degree-attractiveness correlation and smallest when the attractiveness and the popularity are positively correlated.

The non-monotonic behavior of the matching ratio R is also interesting from a stochastic optimization viewpoint: the simple trial-and-error matching procedure, governed and constrained by individuals' attractiveness, fares reasonably well everywhere (against the maximum accessible matching on a given bipartite graph), except for a narrow intermediate thin region (Fig v). The "worst-case" average degree depends strongly on network heterogeneity just non on degree-bewitchery correlations.

Our results revealed the function of topology in the process of human mate choice and can bring further insights into the investigations of different matching processes in different networks [xiii, 34, 37–39]. Indeed, in this work nosotros focused simply on the bones model of the mate seeking process in random networks. However, different variations tin can be considered. For case, there is no degree correlation between the two coupled individuals observed in our model, merely because the networks we studied are random with no assortativity. In reality, the connexion may not be random and and so assortativity tin can be considered. Furthermore, the networks in our model are static and the degree of a node does not alter with time. In reality, a node may proceeds or lose friends and consequently its caste may change. Likewise, stable matching between individuals does non have to last forever, it merely needs to exist an club of magnitude longer than unstable matching. It is possible to establish sure rates to stable matching dissolution and analyze the steady state behavior of so generalized system. Finally, here we considered the attractiveness as a one dimensional aspect of individuals. In more realistic scenarios, attractiveness can be a multi-dimensional variable with different merits [9, xl, 41]. Investigations of such more complicated cases are left to future work.

Funding Statement

This work was supported in role by the Army Research Laboratory under Cooperative Understanding Number W911NF-09-2-0053 and past the Function of Naval Research (ONR) grant no. N00014-09-i-0607. The views and conclusions contained in this certificate are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding whatsoever copyright annotation here on.

Information Availability

All relevant information are within the paper.

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