Strength of minority ties: the role of homophily and group composition in a weighted social network

The original WSN model [12] has been implemented considering different variations, for example, link deletion and aging mechanisms [23], multilayer features [24], and extreme homophily [25]. Here, we introduce an extension that considers nodes with different attributes and continuous attribute-dependent interactions. Based on social network theory [11], the network evolution is governed by the following rules (see figure 1(a)):

• The global attachment (GA) process is controlled by the parameter $p_r\in[0,1]$ which defines the probability of random or focal closure connections. For each node i, a node j (not connected to i) is randomly selected, then an edge of weight $w_{ij} = w_{ji} = w_0$ is created with probability p_r between the two. If the degree of node i is zero, then, an edge of weight $w_{ij} = w_0$ is created with $p_r = 1$.
• The local attachment (LA) process is controlled by the parameter $p_t\in[0,1]$ which defines the probability of local or triadic closure connections. For each node i (with degree different from zero), a node j is selected among its neighbors with probability proportional to the weight w_ij , then another node k is selected among the neighbors of node j with probability proportional to w_jk . Here, three scenarios are possible: (i) if node j has node i as its only neighbor, the weight w_ij is incremented by a quantity δ; (ii) if node k is also neighbor of node i, then, the weights w_ij , w_jk , and w_ki , are incremented by δ; (iii) if node k is not a neighbor of node i, then, an edge of weight $w_{ki} = w_0$ is created with probability p_t and the weights w_ij and w_jk are incremented by δ. The parameter δ hence reinforces the already existing edges and produces heterogeneous weights over time.
• Node/edge deletion. The deletion process is controlled by the parameter $p_d\in[0,1]$ which defines the probability of edge or node deletion. In the case of nodes, a random node i is removed and replaced with a new node (with no connections) with probability p_d . In the case of edge deletion, edges are deleted with probability p_d . In any case, with $p_d = 0$, edges or nodes are permanent.

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We extended the WSN model to account for the homophilic effects of the nodes' attributes over the GA and LA tie-formation processes. This was done by considering nodes with a 'sex' attribute, $\sigma = \{F,M\}$. The GA and LA processes are now governed by an explicit homophily principle that is modulated by a segregation parameter q that takes continuous values in $[0,1]$, establishing the probability of connections between nodes of different sex. Note that this is quite different from binary or extreme homophily in which nodes can only connect if attributes match exactly [25].

The GA and LA processes are modified as follows. In the case of GA, the focal player or node i randomly chooses a candidate node j (not connected with i); then, i and j connect with probability p_r if the same attributes are shared ($\sigma_i = \sigma_j$), otherwise ($\sigma_i \neq \sigma_j$), they connect with a smaller probability $p_r\times q$. As before, edges are created with $w_{ij} = w_0$. In the case of LA, the local search is performed as before if two nodes involved in a connection have the same attribute: a node i chooses a node j among its neighbors with probability proportional to w_ij , then a node k is selected among the neighbors of node j with probability proportional to w_jk . If a pair of nodes does not have the same attributes, the corresponding action is taken with probability q. Namely, edges between nodes of different attributes are reinforced (when selected) with probability q, and triangles are formed with probability $p_t\times q$.

In this way, q → 1 leads to full mixing (original model), while for q → 0 to full segregation, with non-trivial segregation in-between (see figure 1(b)), a feature that is not considered in previous models.

The empirical association network was originally presented in [2]. The data was collected between January 2013 and September 2014 from a well habituated group of spider monkeys living in the Otoch Ma'ax yetel Kooh protected area in Yucatan, Mexico. The group consisted of 23 monkeys (excluding individuals younger than 5 by the end of the study period): 7 males (M) and 16 females (F); in terms of simplified age class: 13 adults (A; age greater than 8 years), 8 subadults (SA; age 5–8 years), and 2 juveniles (J; age 3–5 years) which became subadults in the course of the study period.

Edges represent associations based on aggregated data from scan sampling of spider-monkey subgroups, performed every 20 min. Scan samples comprise records of subgroup composition collected by experienced observers following one subgroup at a time for 4–8 daily hours during 244 days, hence we do not have the full record of all associations. Given the high degree of fission-fusion dynamics of the species, group members are found organized into subgroups which change their size and composition in the course of hours (a subgroup was defined as the set of individuals within 30 meters of another, see [26] for development and [27] for validation of this definition of subgroup). A pair of individuals was considered to be associated if they were recorded in the same subgroup during a sample. The scan samples hence represent approximations to the instantaneous association patterns between individuals, with a temporal resolution of 20 min. The association index that defines the edges' weights is an aggregated quantity over the whole observation period and considers the proportion of scans two individuals were seen in the same subgroup, and takes values from 0 to 1. For comparison purposes with our network model, the association index is re-scaled by its maximum value.

In figure 1(c), we show a visualization of the empirical network along with some network metrics (see appendix). The study of this type of association network in spider monkeys as well as in other animal species have provided evidence that processes akin to the LA and GA rules as used in the WSN model [12] apply to these species as well (e.g. [4, 28–30]).

The network data is generated with the WSN model considering the parameters' values, $p_r\in[0.1,1.0]$, $p_t\in[0.0,1.0]$, and $q\in[0.0,1.0]$, with steps of 0.1 each. For each parameter set $(p_r, p_t, q)$, an ensemble of S = 100 networks is created. Each network starts from an initial set of N disconnected nodes, following the GA and LA rules modulated by q, for a total of T = 100 iterations. This ensures the formation of a high number of connections and sufficient edge reinforcement as in the empirical network. The probability of edge/node deletion is set to zero ($p_d = 0$) to account for no individual loses (deaths or disappearances), nor loses in the count of interactions, similarly to the way the empirical network was constructed. We also set $\delta = w_0 = 1$.

We rely on seven weighted metrics to fit the model: the average strength, $\langle s \rangle$, the average weighted clustering coefficient, C_w , the weighted number of communities, M_w , the weight distribution function, f(w), and the average weight per edge-type, $\langle w \rangle_{FF}$, $\langle w \rangle_{FM}$, $\langle w \rangle_{MM}$ (see appendix for definitions). Unweighted metrics such as the average degree, $\langle k\rangle$, average clustering coefficient, C, and number of unweighted communities, M, are somehow simple as their empirical values are very close to N − 1, 1, and 1, respectively, and easily reproduced by the model. Therefore, we focus on the weighted metrics only, which take non-trivial values about the weighted social interactions.

The statistical fitting of the model is performed for the seven aforementioned metrics considering three tests. For each set of parameters, $(p_r, p_t, q)$, the empirical metrics must fall within a significance range in order to be accepted. Hence, the set of parameters passing all the tests yield networks whose weighted metrics are statistically consistent with the empirical data. The tests are the following:

• Test 1 (T1) for the weighted macro metrics. For each set of parameters, $(p_r, p_t, q)$, the empirical values, $\langle s \rangle$, C_w , and M_w , must fall within the significance range $[\alpha, 1-\alpha]$ of the corresponding statistical distribution, that is, a two-tail test with significance level α = 0.05.
• Test 2 (T2) for the weight distribution function. For each set of parameters, $(p_r, p_t, q)$, we performed a Kolmogorov–Smirnov (KS) test for the simulated weight distributions, ruling out the cases that have a p-value lower than 0.1 [31].
• Test 3 (T3) for the average weight according to edge-type. For each set of parameters, $(p_r, p_t, q)$, the empirical values, $\langle w \rangle_{FF}$, $\langle w \rangle_{FM}$, and $\langle w \rangle_{MM}$, must fall within the significance range $[\alpha, 1-\alpha]$ of the corresponding statistical distribution, that is, a two-tail test with significance level α = 0.05.

Notice that these tests progressively probe the simulated network structures, from general properties (weighted macro metrics), through meso characteristics (weighted distribution function), to more specific features of interest (average weights according to edge-type).

We consider the following features of the empirical network for the simulations and statistical fitting: the number of nodes is set to N = 23, with $N_M = 7$ (males) and $N_F = 16$ (females). Notice that this is not a symmetric composition of the number of males (a minority) relative to the females (the majority group). Also, the nodes include adults (A), subadults (SA), and juveniles (J). In general, the extended WSN model produces non-trivial numerical results for the weighted metrics, for example, in figure S1 (supplementary materials), we show the values for the weighted metrics, $\langle s\rangle$, C_w and M_w , obtained from the model for different projections in parameter space. However, only those sets, $(p_r,p_t,q)$, that satisfy the significance tests are considered. In figure 2, we present the results of a successful test considering the fitted parameters $(p_r,p_t,q) = (0.7,0.5,0.7)$.

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In figure 2(a), the probability distributions of the weighted macro metrics recover the corresponding empirical observations with statistical significance. Recall that the tests are not independent among each other, therefore, these results show that the model is able to reproduce the non-trivial dependency among the average strength of connections, average weighted clustering, and the number of weighted modules, under the male/female composition asymmetry of the empirical network. In addition to recovering the average metrics, after the second test based on KS distances, the model also recovers the empirical weight distribution (see figure 2(b)) which encodes a more detailed structural characteristic of the empirical social structure, namely, the statistical variation in the strength of connections disregarding their type. Remarkably, after the third test based on sex-dependent interactions, we find that our homophilic WSN network model also recovers the average weights according to edge-type, $\langle w \rangle_{FF}$, $\langle w \rangle_{FM}$, and $\langle w \rangle_{MM}$, see figure 2(c).

The fitted parameters in the previous example are just an instance of the various sets that satisfy the three statistical tests. Despite the fact that the nodes of the model do not have an explicit age attribute, in figure 3, we show the distribution of the fitted parameters obtained by gathering different age categories to construct three empirical networks with varying group compositions. Results in figures 3(a) and (b) correspond to the previous association network, with A, SA, and J individuals. In figure 3(a), we present a comparison of the empirical measurements against the simulated results using values close to the averages (assuming a uniform prior) shown in figure 3(b), that is, the values of the distributions that result from the application of all significance tests (T1 + T2 + T3). The joint distributions of the parameters is non-trivial and in figure S2, we also present their distribution in parameter space. Results in figures 3(c) and (d), represent the analysis considering A and SA individuals, with $(p_r,p_t,q) = (0.5,0.5,0.8)$; and in figures 3(e) and (f), exclusively for A individuals, with $(p_r,p_t,q) = (0.2,0.5,0.8)$. Similar to the results in figure 2, in figures S3–S5, we present examples of the statistical fitting for different age groups.

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We interpret the higher strength of male-male connections, observed in figures 3(a) and (c), as a consequence of the difference in the number of males and females in the group (as analyzed in detail in the next subsection). This pattern can be attributed to the effects of the segregation parameter q and the two differentiated groups in the system (F and M), that is, male-male interactions are stronger on average because there are less possible interactions among males than among females. Note that this observation no longer holds when only adult individuals are considered, see figure 3(e). Although the homophilic model still fits the empirical data and recovers network features with statistical significance in that case, the empirical F–F interactions are stronger than M–M ones, despite of the fact that there are slightly more females (7) than males (6). Nonetheless, this makes sense because the majority of subadult females in the empirical system are newly immigrated and, as a result, less integrated into the group (especially with other females), exacerbating the effect of weak relationships among females.

From figure 3, we can notice how the average segregation parameter, $\langle q \rangle$, is well localized for all age compositions with a small standard deviation that validates the strict inequality, $\langle q \rangle\lt 1$, which implies that segregation effects are indeed necessary to reproduce the empirical observations related to sex-dependent interactions for all age compositions; the average triadic closure parameter, $\langle p_t \rangle$, is surprisingly uniformly spread across its all possible values for all age compositions; in contrast the average focal closure parameter, $\langle p_r \rangle$, is well localized but shifts according to the age composition. This finding is explained by looking at the difference in network sizes (as less connections are necessary when nodes of different ages are removed in the different examples) but also to the non-trivial dependency of the parameters p_r and p_t with q (see figure S2, for their distribution in parameter space).

The previous results show that the WSN model with homophily reproduces the empirical evidence regarding stronger male-male relations ($\langle w \rangle_{MM}$) over the other types in the associations of spider monkeys. Furthermore, notice that in such empirical networks there is a asymmetrical group composition, where females represent a majority while males represent a minority group. Thus, in order to have a better understanding of the nature of our results relating to the strength of male-male connections, we considered further numerical explorations. For such simulations we considered a generic bipartisan network with nodes of type F and M, where F-type nodes represent the majority group and M-type nodes the minority. Also, the fitted parameters, $p_r = 1/2$, $p_t = 1/2$, with varying homophily, q, network size, N, and percentage of the minority group, φ_m . Again, no node or edge deletion is considered ($p_d = 0$).

First, to illustrate the effects of the homophily parameter q on the weighted network metrics, we explored different network sizes, $N\in\{20,50,100,200,500\}$, with a fixed percentage of M-type nodes, $\phi_m = 1/2$. As a direct inspection of the results shows (see figure 4(a)), the weighted metrics display characteristic behaviors. The average strength ($\langle s \rangle$) increases monotonically with heterophily (q → 1), rather independently of N. Meanwhile, the behavior of the number of communities (M_w ) seems to indicate a smooth structural crossover controlled by q. Not only M_w is clearly proportional to N, but is also monotonic with respect to q, increasing more sharply around $q = 1/2$. A crossover near $q = 1/2$ is also observed for the average strength, that shows two slope regimes: a nearly constant behavior for $q\lt 1/2$ followed by a linear increase for $q\gt 1/2$ in figure 4(a).

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Furthermore, the weighted clustering coefficient (C_w ) is roughly inversely proportional to N, showing that smaller networks imply tighter groups, as one might expect. However, it is non-monotonic with respect to q, except for small networks (N = 20). Indeed, the behavior of the weighted clustering is not trivial. In the case of large enough networks, a minimal cliquishness is attained for a certain $q\in(0,1)$. But for small networks (N = 20), C_w monotonously increases with heterophily ($q\to 1)$, making the less cliquish networks the ones that are fully sex segregated (q = 0). This behavior is somewhat counter-intuitive since one would expect a network divided into two small connected components (q = 0) to be the most cliquish and reinforced among the other networks (q > 0), where one big connected component is formed but where fewer triangles are closed. We explain this behavior by the fact that at small q, less links per unit time are established and any pair of nodes with different attributes have a higher probability to fail to connect or reinforce their connection. On the other hand, at larger q, the same links have more opportunities to form and be reinforced, as well as triangles. This is specially true for a small network.

Second, we explored in figure 4(b) the effects of q over the weighted network metrics for different percentages of the minority group, $\phi_m = \{0.1,0.2,0.3,0.4,0.5\}$, with fixed network size, N = 500. We chose a large network expecting it would more clearly reveal any macroscopic collective effects likely to emerge. As it can be observed in figure 4(b), we found that the weighted macro metrics also display non-trivial behaviors as the group composition and segregation parameters are varied: the average strength $\langle s \rangle$ increases with q, rather independently of φ_m ; the weighted clustering C_w is non monotonic with q and exhibits a minimum, meaning that the networks are more cliquish when fully segregated (q = 0) or completely mixed (q = 1) than when q ≈ 0.5 (for $\phi_m\gt 0.3$). Hence, an intermediate level of segregation minimizes cliquishness. Noteworthy, the minimum of the weighted clustering (C_w ) at an intermediate value of q might be interpreted as a sign of structural transition, similar to the nontrivial behavior found for the binary clustering coefficient (C) in a related network model with homophily (see figure 2(b) in [25]). In the case of [25], such transition occurs at a specific number of node attributes or features (F), with a maximum in C that increases as the number of feature values increases. In our model, a smooth structural crossover occurs at low or moderate values of the segregation parameter q, where C_w is minimum and decreases as the proportion of the minority group decreases. Although C is a measure of triangle formation while C_w measures the reinforcement of such triangles in weighted networks, these extrema might be interpreted as signs of crossover transitions.

As already noticed in figure 4(a), the number of communities (M_w ) also exhibits an interesting crossover transition with q in figure 4(b). When the percentage of the minority group is small ($\phi_m = 0.1$), the contribution of the latter to the community structure is negligible regardless of q. However, when φ_m is not so small ($\phi_m\gt 0.3$), the number of communities increase sharply with q near q = 0.5, before reaching a plateau independent of q for q > 0.5. We interpret this increase of M_w by the fact that at large q, each individual has effectively more possibilities of connections (outside the set of nodes sharing its attribute). We observe that during a same time period, two separated smaller groups (at q = 0) form overall a smaller number of communities than the full group (q = 1), indicating that M_w is not additive. Together with the previous results for the weighted clustering C_w , the behavior of M_w provides further evidence of a structural crossover regulated by the homophily parameter q.

Finally, in order to test the robustness of the attribute-dependent association strengths, we proceeded to systematically explore the parameter space considering the sizes $N = \{20,50,100,200,500\}$, composition with $\phi_m = \{0.1,0.2,0.3,0.4,0.5\}$, and segregation effects with $q = \{0.1,0.3,0.5,0.7,1.0\}$. In figure 5(a), we show the average weights per edge-type in parameter space. These average values vary according to the network size and display greater asymmetries for high segregation (q → 0) than for homogeneous or fully mixed networks (q → 1). It can be observed that in sex-dependent weight averages, $\langle w \rangle_{FF}$, $\langle w \rangle_{FM}$ and $\langle w \rangle_{MM}$, segregation and asymmetries in group composition tend to favor strong interactions among minorities, in this case, $\langle w \rangle_{MM}$. The interaction strength among members of the majority, $\langle w \rangle_{FF}$, vary relatively little with the segregation parameter q. As expected, the $\langle w \rangle_{FM}$ interactions increase with q but it is quite insensitive to the composition. In contrast to $\langle w \rangle_{FF}$, the strength of minority interactions $\langle w \rangle_{MM}$ increases as segregation becomes stronger, mostly for small minorities. For a clearer visualization, in figure 5(b), the metrics have been divided by a normalization factor such that, for each case, $\langle w \rangle_{FF} + \langle w \rangle_{FM} + \langle w\rangle_{MM} = 1$. This normalization confirms that, indeed, the tendency to favor strong interactions among minorities (male-male interactions, $\langle w \rangle_{MM}$) is a general effect that solely depends on segregation and node-type composition (φ_m ) and homophily conditions (q), independently of the network size (N).

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