Exploring network structure and central items of the Narcissistic Personality Inventory

Abstract Objectives The aim of this work is to explore the Narcissistic Personality Inventory (NPI) using network analysis in a dataset of 942 university students from the French‐speaking part of Belgium. Methods We estimated an Ising Model for the forty items in the questionnaire and explored item interconnectedness with strength centrality. We provide in the supplementary materials the dataset used for the analyses as well as the full code to ensure the reproducibility of our results. Results The NPI is presented as an overall positively connected network with items from entitlement, authority and superiority reporting the highest centrality estimates. Conclusions Network analysis highlights new properties of items from the NPI. Future studies should endeavor to replicate our findings in other samples, both clinical and non‐clinical.

other items in the questionnaire, including items from different domains, such as items 12 ("I like to have authority over other people") and 27 ("I have a strong will to power") that respectively belong to authority and entitlement. In a network structure, we would expect these items to be strongly associated.
It is plausible to consider narcissism as a network of components (in this case, items from a self-reported questionnaire indicating an individual's perspective on narcissistic traits) that mutually influence each other instead of being passive consequences of the same construct. The network approach to narcissism is relevant because it might allow in clinical samples the identification of meaningful targets for intervention, even more so if considered that normal and pathological narcissism form a continuum.
The aim of this work is to explore for the first time NPI items and their relationship in a network of narcissism, therefore applying network analysis to the items of the questionnaire. Network analysis has been shown to offer substantial insight as a complementary tool to factor analysis, which is a more established technique in the field of personality assessment (Briganti et al., 2018): as mentioned, modeling a construct or mental disorder as a network can highlight connections between items or symptoms which can therefore be used for intervention (Blanken et al., 2019).
First, we want to explore the connectivity of the NPI network.
Second, we want to explore the importance of each item in the questionnaire using strength centrality, which is the absolute sum of connections of a given node in the network (Boccaletti, Latora, Moreno, Chavez, & Hwang, 2006).

| Participants
The dataset used for this study is composed of 942 university students from the French-speaking region of Belgium. The participants were first-year students in several Belgian universities and in different undergraduate courses and they volunteered to fill a set of questionnaires which included a French version of the NPI among other questionnaires such as the Interpersonal Reactivity Index (Briganti et al., 2018), the Contingencies of Self-Worth Scale , the Resilience Scale for Adults , the Zung Depression Scale and the Toronto Alexithymia Scale. The questionnaire of the French version of the NPI is fully detailed in a previous paper (Braun, Kempenaers, Linkowski, & Loas, 2016).

| Network estimation
An Ising Model (IM) was estimated from our dataset. An IM (Marsman et al., 2018;van Borkulo et al., 2014) is the binary equivalent of the Gaussian Graphical Model used for continuous datasets (Epskamp et al., 2017). A lasso (least absolute shrinkage and selection operator) was used to provide a conservative network structure (Epskamp & Fried, 2018). We used the default eLasso procedure which combines gamma (to select how many edges the model recovers) was set by default at 0.25; the optimal tuning parameter lambda (used to select the model with the best fit) was automatically chosen by the eLasso procedure. The network structure resulting from this estimation contains items from the NPI represented as nodes. An edge is a connection between two nodes in the network, which is interpreted as the existence of a connection between two nodes controlling for all other nodes in the network.
While estimating a network structure from items of a questionnaire, a connection between two nodes means that the observed group answers on average in a similar way to both items of the questionnaire (Briganti et al., 2018). Each edge in the network represents either a positive (visualized as blue edges) or a negative connection (visualized as red edges). The thickness and color saturation of an edge denotes its weight (the strength of the connection between two nodes).
The Fruchterman-Reingold algorithm places the items in the network based on the inverse of the sum of connections of a given node with other nodes (Fruchterman & Reingold, 1991): this means that strongly connected nodes are put closer in the network visualization.

| Network inference
We estimated strength centrality (Boccaletti et al., 2006) for the 40 items in the questionnaire. Strength centrality represents the absolute sum of the edges of a given node and therefore informs us of the connectedness of items in the network (Briganti et al., 2018).

| Network stability
Stability analyses (Epskamp et al., 2017) were carried out through bootstrapping, which is a repeated estimation of a model under sampled data: we used 2000 bootstraps in this paper. An edge weight difference test was performed to compare all edges against all other edges and to answer the question "is edge A significantly stronger than edge B?". Centrality stability analyses for strength centrality were also carried out to answer the question "is the centrality order stable?". Centrality difference test was performed to answer the question "is the centrality estimate of node A statistically different from that of node B?" We used the subsetting bootstrap procedure that re-estimates the network with a dropping percentage of participants to determine the stability of centrality estimation, and results in a centrality-stability coefficient (CS-coefficient) that should not be lower than 0.25 and preferably above 0.5.
Both difference tests (edge weight and centrality) are carried out by estimating confidence intervals around the difference of two ele-
The average NPI score of the participants of this study was 13 (out of 40), and the standard deviation was 6.4. to the same domain of narcissism: for instance, item 10 ("I see myself as a good leader") is strongly associated to item 33 ("I would prefer to be a leader") and both belong to the authority domain; item 7 ("I like to be the center of attention") presents the second strongest connection in the network to item 30 ("I really like to be the center of attention") and both belong to exhibitionism; item 9 ("I think I am a special person")

| Network of narcissism
shares the strongest edge of the network with item 40 ("I think I am an extraordinary person") and both belong to the superiority cluster. In the case of these three connections, the items involved in an edge measure the same aspect of the construct.
Several connections are found between items belonging to different domains, and we want to illustrate some of these connections.
Domains superiority and self-sufficiency are connected through items 9 ("I think I am a special person") and 39 ("I am more capable than other people"); domains authority and entitlement connect through items 12 ("I like to have authority over other people") and 27 ("I have a strong will to power"); domains authority and exploitativeness connect through items 1 ("I have a natural talent for influencing people") and 35 ("I can make anybody believe anything I want them to"). These domains also tend to measure the same thing, even though belonging to different domains. Some small, negative edge are also found in the network, such as the one between items 11 ("I am assertive") and 24 ("I expect a great deal from other people"). Figure 2 shows strength centrality estimates for the 40-item NPI. Item 27 from entitlement ("I have a strong will to power") presents the highest strength estimate, which means that it is the most interconnected node in the network. Other strong items include item 33 from authority ("I would prefer to be a leader") and item 40 from superiority ("I am an extraordinary person"). Several items present with a strength centrality of 0, which means that they are not connected with any item in the network.

| Network stability
The edge weight bootstrap shows relatively narrow CIs, which indicates a precise estimation of the edge weights in the network. The edge-weight difference test performed shows that stronger edges are significantly stronger than other edges in the network; however, edges 9-40 and 7-30 are not statistically different from each other, which means that, even though edge 9-40 reports a stronger connection in the network, we cannot safely interpret it to be statistically stronger than edge 7-30.
Strength centrality stability analyses report that the centrality order is relatively stable, with a centrality stability coefficient (CScoefficient) of 0.67 (for more information, see Briganti et al., 2018).
Strength centrality difference test reports that stronger centrality estimates are significantly stronger than other estimates but are not significantly different from each other; for instance, we cannot infer whether the centrality of item 27 is really stronger than that of item 33. We obtained a CS-coefficient of 0.67, which indicates stable results.
F I G U R E 1 Network composed of the 40 items from the NPI (details in Table 1). Each item is represented by a node (1 to 40) and belongs to a different domain of the NPI (indicated by a color code). The name of each node is composed as following: an abbreviation of the domain to which the item belongs to followed by the item number F I G U R E 2 Strength centrality estimates for the 40 items of the NPI. The Y-axis represents centrality indices (the higher the estimate the more central the item), and the X-axis represents the 40 NPI items