Table 1.
Definitions of social influence metrics (i.e., betweenness centrality or age category) network level indices (weighted (W) diameter, global efficiency and modularity, as well as unweighted clustering coefficient) along with formulas we used to calculate them; and the expected outcomes per deletion proportion ranging from 0 to 0.2 in increments of 0.04. and type (i.e., targeted or random).
The impact of deletions on each network level index was measured after incremental deletion of the most socially influential individuals while targeting individuals with high betweenness centrality or age category, or when individuals were deleted at random. Our expectations are expressed with a greater- or less-than sign (> or <). For instance, we predicted that relative to random deletion, targeted deletion of seniors would result in lower clustering coefficient values; and that higher deletion proportions would also result in lower clustering coefficient values. (1). Our procedure assumes that the higher the weight of a link between two individuals (or nodes), the shorter the distance between them. To reflect this relationship, we define the length of a link as the inverse of its weight. Using the inverse of the weights of the links connecting all pairs of nodes, we calculated all shortest paths in the network [50,97]. (2). Social transfer is a theoretical expression of the efficiency of passing of transmissible currency, such as information, assumed to be diffusing across network links [47].
Fig 1.
The distribution of association indices in (A) the empirically based versus (B) virtual populations, as a function of age category and kinship of the associating individuals. Age categories are abbreviated using the following symbols: Y—young adult; P—prime adult; M—mature adult; G–the matriarch. During each random deletion, the same proportion of individuals as in targeted deletions was removed randomly. After every deletion proportion, we recalculated the following network level indices: clustering coefficient, as well as weighted diameter, global efficiency and modularity (Table 1). As in the empirically based portion of our study, we used the Hedge’s g test to quantify the difference in the effect size between the means of all network indices across 1) the deletion proportion spectrum, 2) deletion type and 3) deletion metric [96].
Fig 2.
Flow chart summarizing the process of simulating social networks among virtual elephant populations.
At initialization, the probabilities of association within and between groups are set according to kinship and age category (Fig 1B). At the beginning of each time step, the set probability of association within each group or between each set of groups, and between each dyad, is compared to a randomly generated number (RDN) between {0,1}. If this probability is greater than RDN, the association is set to occur. If this probability is lower than RDN, the association does not occur, and the time step is terminated. At the end of each time step the number of times a specific dyad has formed across all previous time steps is updated (i.e., increased by one if the association had occurred, or remained the same otherwise).
Fig 3.
Social network graphs of the empirically based population with color partitioning according to a core group, considered from the perspective of either (A) age category or (B) betweenness centrality; and a comparable example of a virtual population with the partitioning according to a core group, and either (C) age category or (D) betweenness centrality. The nodes are ranked by size where the largest nodes indicate oldest age or highest betweenness centrality. The links are color coded to match the nodes they originate from and ranked according to their relative weight. The thickness scheme depicting the weight of each link ranges from thin (low) to thick (high weight). The links with weight less than 5 percent were filtered out for visual clarity.
Fig 4.
Graphs representing results (mean plus 95% confidence interval) of 100 deletions per each combination of deletion proportion (i.e., 0–20%) and type (i.e., random vs. targeted) in the empirically based network.
The deletions were either targeted according to age category (black series) or betweenness centrality (blue series); or were random (grey and teal series represent random deletions without considering individual traits conducted as control conditions to age- or betweenness centrality-targeted experiments, respectively). The network indices evaluated included clustering coefficient as well as weighted modularity, diameter and global efficiency. For a cross-species context, the minima of y-axis ranges per clustering coefficient as well as weighted modularity and global efficiency are plotted to express the minima from a similar, theoretical treatment in an egalitarian primate society [46]. The weighted diameter index depends on group size, thus the pertinent y-axis is not expressed in a cross-species context. For results of Hedge’s g test expressing the difference in the effect size between the means of each network index between targeted versus random deletions along the deletion proportion axis and per deletion type, refer to S2 Table.
Fig 5.
Graphs representing results (mean plus 95% confidence interval) of 100 deletions per each combination of deletion proportion (i.e., 0–20%) and type (i.e., random vs. targeted) in an example virtual network that is comparable in size to the empirically based social network (see Figs 3 and 4 for detail).
The deletions were either targeted according to age category (black series) or betweenness centrality (blue series); or were random (grey and teal series represent random deletions without considering individual traits conducted as control conditions to age- or betweenness centrality-targeted experiments, respectively). The network indices evaluated included clustering coefficient as well as weighted modularity, diameter and global efficiency. For a cross-species context, the minima of y-axis ranges per clustering coefficient as well as weighted modularity and global efficiency are plotted to express the minima from a similar, theoretical treatment in an egalitarian primate society [46]. The weighted diameter index depends on group size, thus the pertinent y-axis is not expressed in a cross-species context. For results of Hedge’s g test expressing the difference in the effect size between the means of each network index between targeted versus random deletions along the deletion proportion axis and per deletion type, refer to S3 Table.