The Enmity Paradox

It is a fact of mathematics that, in a population with variance in the degree distribution (and subject to certain provisos), the friendship paradox exists. But its extent is very much a result of underlying social factors (such as variation across people in the number of friends they want, or whether popular people are preferred as friends). Some aspects of this phenomenon can also be self-reinforcing as a consequence of positive feedback that results from the biased perception [16].

However, even though there is an intrinsic mathematical symmetry between positive and negative network objects (as instantiated by adjacency matrices), the empirical existence of an enmity paradox is not assured. We mathematically clarify and then empirically investigate the existence, origins, and manifestations of the enmity paradox.

First, we derive the equations for the enmity paradox, which are similar to the equations derived for the friendship paradox [1]. Based on our comparison of the mean number of enemies of individuals with the mean number of enemies’ enemies in the negative world, we theoretically assert that the enmity paradox should indeed arise in a similar manner to the friendship paradox.

Consider a simple signed network $G=(V,E_{(+)},E_{(-)})$, where $V$ is the vertex set with $n$ nodes, and $E_{(+)}$ and $E_{(-)}$ are the set of positive and negative edges revealing two not-necessarily-dependent worlds. The adjacency matrices corresponding to the positive and negative interactions are denoted as $A_{(+)}$ and $A_{(-)}$, respectively. In order to create undirected networks, we either (1) remove unreciprocated edges from the network, or (2) symmetrize the network by removing the edges’ direction. For example, the node $j$ is considered a neighbor of the node $i$ if, in the former, both edges $(i,j)$ and $(j,i)$ are included in the set $E$, while, in the latter, if at least one of these edges exist in the edge set. An edge is referred to as a “friend” if it belongs to $E_{(+)}$, and as an “enemy” if it belongs to $E_{(-)}$. In addition, we also refer to a node $j$ as an $\ell$-hop neighbor of node $i$ if there is a walk with length $\ell$ between $i$ and $j$; a walk can include repeated nodes. Here, we focus solely on the enmity and friendship paradoxes for undirected networks. The enmity paradox for directed networks are provided in the Appendix D.

There are two types of degrees for each node $i$ in these two parallel worlds, $k_{(+),i}=\sum_{j}A_{(+),ij}$ corresponding to positive network, and $k_{(-),i}=\sum_{j}A_{(-),ij}$ corresponding to negative network. The probability of a node with degree $k_{(+)}$ ($k_{(-)}$) is denoted as $p_{k_{(+)}}^{(0)}$ ($p_{k_{(-)}}^{(0)}$) or, more simply, $p_{k_{(+)}}$ ($p_{k_{(-)}}$); and the probability of a node’s friend with positive degree $k_{(+)}$ and negative degree $k_{(-)}$ is denoted as $p_{k_{(+)}}^{(1)}$ and $p_{k_{(-)}}^{(1)}$, respectively. Similarly, the probability of a node’s enemy with positive degree $k_{(+)}$ and negative degree $k_{(-)}$ is denoted as $q_{k_{(+)}}^{(1)}$ and $q_{k_{(-)}}^{(1)}$, respectively. For simplicity, whenever it is clear from context, we denote the degree, the degree distribution, and the degree distributions of neighboring nodes corresponding to the friendship and enmity networks using $k$, $p_{k}$, and $p_{k}^{(1)}$ and $q_{k}^{(1)}$.

To establish the enmity paradox, we need to compare the average negative degree of a random node with the average negative degree of a random enemy of a random node. The average negative degree of a random node can be written as $\langle k_{(-)}\rangle_{p_{k_{(-)}}}=\sum_{k_{(-)}}k_{(-)}p_{k_{(-)}}$. Going along a random negative edge to one of its enemies leads to a node with negative degree $k_{(-)}$ with probability $q_{k_{(-)}}^{(1)}$ proportional to $k_{(-)}p_{k_{(-)}}$, i.e., $q_{k_{(-)}}^{(1)}=\nicefrac{{k_{(-)}p_{k_{(-)}}}}{{\langle k_{(-)}\rangle_{p_{k_{(-)}}}}}$. Therefore, the average degree of a random enemy can be written as $\langle k_{(-)}\rangle_{q_{k_{(-)}}^{(1)}}=\sum_{k_{(-)}}k_{(-)}q_{k_{(-)}}^{(1)}=\sum_{k_{(-)}}\nicefrac{{{k_{(-)}}^{2}p_{k_{(-)}}}}{{\langle k_{(-)}\rangle_{p_{k_{(-)}}}}}$ for enmity networks. Using Jenson’s inequality, it can be shown that $\langle k_{(-)}\rangle_{q_{k_{(-)}}^{(1)}}\geq\langle k_{(-)}\rangle_{p_{k_{(-)}}}$. This inequality can be called the “enmity paradox” for the negative world, and it follows from the same mathematical facts as the friendship paradox in the positive world.

Social networks that simultaneously involve both positive and negative ties are theoretically more complicated. That is, analytically, we have shown that people have fewer enemies on average than their enemies; however, the result of mixing positive and negative worlds is not obvious. Do people have more or fewer enemies than their friends, or do they have more or fewer friends than their enemies?

Considering a dependency between positive and negative degrees with the correlation denoted by $\rho_{k_{(+)},k_{(-)}}=\nicefrac{{\left(E\left[{k_{(+)}k_{(-)}}\right]-E{k_{(+)}}E{k_{(-)}}\right)}}{{\sigma_{k_{(+)}}\sigma_{k_{(-)}}}}$, we have the joint probability of degrees $k_{(+)}$ and $k_{(-)}$ as $p_{k_{(+)},k_{(-)}}=f(\sigma_{k_{(+)},k_{(-)}})$. For this scenario, the average degree of a person’s enemies would be denoted by $\langle k_{(-)}\rangle_{p_{k_{(-)}}}$; the average degree of a person’s friends would be denoted by $\langle k_{(+)}\rangle_{p_{k_{(+)}}}$; the average degree of a person’s friends’ enemies would be denoted by $\langle k_{(-)}\rangle_{p_{k_{(+)},k_{(-)}}^{(1)}}$; and the average degree of a person’s enemies’ friends would be denoted by $\langle k_{(+)}\rangle_{q_{k_{(+)},k_{(-)}}^{(1)}}$, where the first two are driven before and the last two are as follows.

Therefore, we have three different regimes as follows: (1) If we have independent positive and negative worlds, i.e., $p_{k_{(+)},k_{(-)}}=p_{k_{(+)}}p_{k_{(-)}}$, then $\langle k_{(-)}\rangle_{p_{k_{(+)},k_{(-)}}^{(1)}}=\langle k_{(-)}\rangle_{p_{k_{(-)}}}$ and there is no difference between the average number of a person’s enemies and average number of enemies of a person’s friends, and similarly $\langle k_{(+)}\rangle_{q_{k_{(+)},k_{(-)}}^{(1)}}=\langle k_{(+)}\rangle_{p_{k_{(+)}}}$, which means no difference between the average number of a person’s friends and the average number of a person’s enemies’ friends. (2) If there is a positive correlation between the two worlds, i.e., $\rho_{k_{(+)},k_{(-)}}>0$ (or when $\sigma_{k_{(+)},k_{(-)}}>0$), then we have paradoxes in the mixed world in the same direction as the enmity and friendship paradoxes, as a person’s friends have more enemies and a person’s enemies have more friends compared to a person. (3) Finally, if there is a negative correlation between the two worlds, i.e., $\rho_{k_{(+)},k_{(-)}}<0$ (or when $\sigma_{k_{(+)},k_{(-)}}<0$), then we have paradoxes in the mixed world in the opposite direction with the enmity and friendship paradoxes, as a person’s friends have a smaller number of enemies and a person’s enemies have a smaller number of friends compared to the person.

In this section, we use a similar representation to the one used in the previous section to explore when the enmity paradox can be generalized for non-topological attributes such as happiness, wealth, and health (e.g., on average, a person’s friends are richer or happier than they are, as has previously been shown). Due to the similarity between the enmity and friendship paradox, we expect that the condition for generalization of the global definition is similar to the condition previously studied for the generalized friendship paradox [2]. Similarly, the condition for the relationship between the local and global formulation of the generalized enmity paradox should be similar to the condition of their relationships for the generalized friendship paradox [6]. Given the vector of non-topological characteristic denoted by $x$, we define a diagonal matrix $D_{x}$ with diagonal entries $x_{i},i\in\{1,...,n\}$. Thus, the generalized enmity paradox for the global definition of the difference $\delta_{g,-w}(x)$ can be written as Eq. 8.

The numerator can be reduced to $\operatorname{Tr}{\left[D_{(-)}\left(\mathbb{J}D_{x}\mathbb{J}-D_{x}\mathbb{J}^{2}\right)\right]}=\mathbf{1}^{T}D_{(-)}(J-n\mathbb{I})D_{x}\mathbf{1}=\sum_{i,j}k_{(-),i}x_{j}-n\sum_{i}k_{(-),i}\sum_{i}x_{i}$, which is negative only if there is a negative correlation between $k$ and $x$.

The local definition of enmity paradox can also be written as Eq. Generalized enmity paradox.

Using $\mathbb{I}=D_{(-)}^{-1}D_{(-)}$ and $D_{(-)}\mathbf{1}=A_{(-)}\mathbf{1}$, the numerator can be reduced to

Here, there are a variety of possibilities; for example, if $x_{i}<k_{i}$, this quantity is always negative and all local differences are also negative, then this is merely an uninteresting sufficient condition. The relationship between $\delta_{g,-w}(x)$ and $\delta_{l,-w}(x)$ can be formalized using the edge-based correlation between $x_{i}$ on one endpoint of edge $(i,j)$ and the inverse degree of another endpoint of that edge, $\nicefrac{{1}}{{k_{(-),j}}}$, i.e., $\delta_{g,-w}(x)-\delta_{l,-w}(x)\propto\rho_{(x)}$, where $\rho_{(x)}={\rm{cor}}(x_{i},\nicefrac{{1}}{{k_{(-),j}}}|(i,j)\in E_{(-)})$ [6]. If the aforementioned edge-based correlation is positive, we have $\delta_{l,-w}(x)<\delta_{g,-w}(x)$; if it is negative, we have $\delta_{g,-w}(x)<\delta_{l,-w}(x)$; and if there is no correlation, then two measures are equivalent. Using this relationship we have four possibilities of (positive, positive), (positive, negative), (negative, positive), and (negative, negative) for ($\delta_{l,-w}(x)$, $\delta_{g,-w}(x)$) differences.

We can also consider the paradoxes for higher-order neighbors. For example, in the global formulation, if the average degree of a random neighbor of a random node is greater than the average degree of a random node, is it also the case that the average degree of a random $2$-hop neighbor of a random node is also larger—or for any random $\ell$-hop neighbor of a random node (an $\ell$-hop neighbor is a neighbor we arrive through a random walk with length $\ell$).

In this case, it is more specific to ask if there is a relationship between the paradox strength and the order of the neighbors. Here, the equation for global formulation of $\delta_{g}$ for an order of $\ell$ can be written as Eq. 10,

where, for $\ell=2$ can be written as follows.

Accordingly, we can see different scenarios under different conditions. If the degree of nodes that are connected have a positive correlation, i.e., $\nicefrac{{1}}{{n}}\sum_{i\sim j}d_{i}d_{j}>\nicefrac{{1}}{{n^{2}}}\sum_{i}d_{i}\sum_{j}d_{j}^{2}$, we still have similar paradoxes, i.e., the average degree of a neighbor of a neighbor of a random node is larger than the average degree of a random node. If we have the opposite inequality, we see a paradox in a counterintuitive sense in comparison with the friendship paradox. And if we have equality, we do not see any paradox. However, the paradox is always valid for $\ell$ odd since the numerator of Eq. 10 is always negative for $\ell$ odd due to Theorem 4.2 in Ref. [5] that was originally proposed in Ref. [17]. The theorem says that, given positive integers $r$ and $s$ such that $r+s$ is even, we have $\mathbf{1}^{T}A^{r}\mathbf{1}\cdot\mathbf{1}^{T}A^{s}\mathbf{1}\leq\mathbf{1}^{T}A^{r+s}\mathbf{1}\cdot\mathbf{1}^{T}\mathbf{1}$.

For a local definition, the equation can be reduced to Eq. 12,

where it can be both positive or negative under different circumstances. Statements similar to the relationship of $\delta_{l}$ and $\delta_{g}$ can be derived for this purpose.

Hence, theory suggests that the enmity and friendship paradoxes for higher orders may not always be true and might only hold under some circumstances (e.g., particular sorts of networks, or particular regimes such as related to whether geodesic backsteps are allowed); while we note these mathematical observations here, we leave empirical investigation of these details to future work.

Despite the mathematical similarity of the enmity paradox formulation and the friendship paradox, the empirical existence of the enmity paradox in real networks is not guaranteed given the differences between the negative and positive environments. Positivity, for example, is characterized by a large clustering coefficient, reciprocity, and homophily [18, 19], while negativity can even seem like mere noise [18, 20]. Here, we characterize some of these structural characteristics of social networks before investigating the enmity paradox in comparison between the positive and negative ties. These noteworthy differences can be partly explained via the inherent avoidance between the sender and receiver of negative ties, which leads to low information transfer [18]; in other words, people are more likely to know who likes them (because the other persons are more likely to so declare) than they are to know who dislikes them (which is information that is more often kept private) [21].

Several distinct measures might play an important role in explaining the friendship paradox. (1) We denote the clustering coefficient using $T_{l}$, which is the local adaptation of the transitivity measure (Appendix B), and transitivity as $T_{g}$. (2) The inversity measure ($H_{i}$) (which is not the same as degree assortativity, with which is negatively correlated) has a direct effect on the local friendship paradox strength [12]. (3) The degree assortativity ($H_{a}$) is also important [14]. (4) A heterogeneity index that captures starlike graphs—similar to the inversity measure—also seems to play an important role for the severity of friendship paradox. For degree assortativity, we can use extant measures [22]; and for starlike strength, we use a measure that represents topological heterogeneity in complex networks and is maximal for star graphs ($H_{\ast}$) [23]. We also consider (5) the variance of the degree ($H_{\rm{var}}$), and (6) a novel heterogeneity measure ($H_{\rm{deg-div}}$) that reflects degree diversity and has a close relationship with entropy [24]. To characterize other differences between positive and negative ties, we can also compare four more topological properties, including the reciprocity, the clustering coefficient, the homophily between the positive and negative ties, and the normalized betweenness-centrality (the mathematical definitions for all these measures are provided in Appendix B).

We use data from a sociocentric network study of 24,687 people aged 11 to 93 years (with a mean age of 32) in 176 geographically isolated villages in western Honduras [11]. We first construct 176 binary directed signed networks. We use three name generators to determine (potentially overlapping) positive ties (“Who do you spend your free time with?” “Who is your closest friend?” and “Who do you discuss personal matters with?”). And we use one name generator for negative ties (“Who are the people with whom you do not get along well?”). Here, we focus on the enmity paradox for undirected (symmetrized) networks (an example village is illustrated in Fig. 2). Results for undirected (reciprocated) and directed networks are in the Appendices C and D, respectively.

Across the whole dataset, nodes have an average of $6.89$ (${\rm{SD}}=3.79$) friends, and friends have an average of $8.97$ (${\rm{SD}}=4.4$) friends. In order to find the difference at the individual level, we must remove isolated nodes; and the average consequent difference is $-1.5$ (${\rm{SD}}=3.57$). For the enmity networks, a node has an average of $1.26$ (${\rm{SD}}=1.70$) enemies, while an enemy has an average of $3.55$ (${\rm{SD}}=2.58$) enemies. After removing the isolated nodes, the average difference between the number of enemies and the number of enemies of enemies is $-1.10$ (${\rm{SD}}=2.57$). For the remaining analyses, we remove the isolated nodes from the constructed networks.

Core topological properties of friendship and enmity networks are presented in Fig. 3. The reciprocity of positive edges is much larger than the reciprocity of negative edges, in part due to the relative scarcity of negative ties. Therefore, we expect the undirected enmity networks constructed from reciprocated edges to be much sparser compared to the friendship networks (Appendix C). The clustering coefficient of positive edges is much larger than the clustering coefficient of negative edges; that is, friendship networks have more triangles than enmity networks (Fig. 2). The presence of starlike motifs is therefore expected to be more prevalent in negative environments (Fig. 2). This is in turn aligned with the results for both $H_{\ast}$, and $H_{i}$. The variance of degree for positive networks is much larger than the negative networks. Thus, it appears that, among the different factors contributing to the strength of the enmity paradox, the starlike indices ($H_{i}$ and $H_{\ast}$) align with greater strength, and the lower $H_{\rm{var}}$ opposes it. Finally, we also plotted the difference between the normalized betweenness centrality (${\rm{N-BC}}$) of the positive and negative worlds to emphasize how the negative world has a more starlike shape as the maximum value of unnormalized ${\rm{BC}}$ is achieved by the central point in a star network [25].

First, we study the global and local enmity paradox among the 176 villages and we present the histogram of these values in Fig. 4. We observe very similar strengths in the enmity and friendship paradoxes in negative and positive worlds, respectively. Additionally, given slightly negative $H_{i}$ values for friendship networks, we expect the strength of local paradox to be greater than the global paradox for friendship networks as compared to enmity networks (Eq. Computation and representation of enmity paradox for empirical data, Fig. 4, A and D, and Appendix, Fig. S12). The global and local paradoxes for enmity networks and also for the mixed worlds, however, are nearly equivalent, as expected due to their balanced $H_{i}$ values (Appendix, Fig. S12).

The results for the mixed worlds are also presented in Fig. 4. Although the paradoxes almost disappear in undirected (reciprocated) networks due to the sparsity of remaining negative edges (Appendix, Fig. S1), for undirected (symmetrized) networks, there is comparable strength in the paradoxes. In other words, in undirected networks using symmetrized edges, our friends are more likely to have enemies than we are (Fig. 4B) and our enemies are more likely to have friends than we are (Fig. 4C).

To understand the effect of different topological properties noted in Fig. 3 on the enmity paradox strength, we analyze the relationship between the local and global paradox measures and the topological properties. Using regression models, we can characterize the effect of various topological features on friendship and enmity paradox strengths (Appendix E). Among the measures, the degree variance $H_{\rm{var}}$ and the starlike embedding $H_{\ast}$, besides the degree diversity $H_{\rm{deg-div}}$ and inversity $H_{i}$, can explain a significant portion of the intervillage variance in paradox strengths ($R^{2}\in[0.94,0.97]$). Furthermore, $H_{\rm{var}}$ and $H_{\ast}$ also have large effects. This relationship is illustrated in Fig. 5 (the relationships between the strength and other heterogeneity measures are highlighted in Appendix, Figs. S5 and S6). As expected, the larger the variance and the more the starlike embedding, the stronger the paradox.

The higher-order enmity and friendship paradoxes are presented in Fig. 6 (for the first to the sixth order). Because high-degree individuals are counted exponentially more than individuals with low degrees when increasing the order of walks, the higher-order enmity and friendship paradoxes are more severe than the typical enmity and friendship paradoxes. This phenomenon is much more severe in local paradoxes when we compare the degree of a random node with its higher-order neighbors (Fig. 6, bottom row).

To investigate the relationship between a node’s contribution in the local enmity and friendship paradoxes and its topological features, we plot the nodal contribution versus the location of the nodes in a network. The location of the nodes is defined by using a simple fact in networks, namely that the nodes in the center of a network have a smaller distance from other nodes while the nodes in the periphery have a larger distance from other nodes. Therefore, if we embed every node using its average and standard deviation of distances from other nodes in a network—normalized by the network diameter within each network component separately—we find that, in a network, peripheral nodes have a greater paradox strength than central nodes (Fig. 7).

Finally, the results for the generalized enmity paradox are provided in Appendix F, Tables S3, and S4, and Figs. S8 and S9. The results document the existence, for instance, of the generalized paradoxes in both enmity and friendship with respect to wealth. Due to the higher correlation between non-topological features and positive degree, the generalized friendship paradox is stronger than the generalized enmity paradox.

The results for undirected (reciprocated) and directed networks are provided in Appendix, Figs. S1 and S3, respectively. For undirected (reciprocated) networks, the enmity and mixed-world paradox strengths are much smaller compared to undirected (symmetrized) networks—due to the small reciprocity in enmity networks. For directed networks we almost have larger global enmity paradox strength. The local enmity and mixed-world paradox strengths can be explained using an inversity measure [6] (Appendices D and G, and Fig. S13).

Our perception of the world can be distorted by the sampling bias towards high-degree nodes that arise as a result of the friendship paradox [1]. In our minds, if not in reality, our friends have more friends, our collaborators have more collaborators, and our colleagues are wealthier or happier than we ourselves. Here, we showed that the same observations hold with respect to antagonistic ties. Yet, with a lower clustering coefficient and lower degree assortativity and with greater inversity, we expect even more starlike shapes in enmity networks. These factors all make the enmity paradox even more probable than the friendship paradox. Moreover, the enmity paradox obtains in a negative world when comparing the number of an ego’s enemies with their enemies’ enemies as well as in mixed worlds when we compare an ego’s friends with the ego’s enemies’ friends and an ego’s enemies with an ego’s friends’ enemies.

Some topological differences between enmity and friendship networks (e.g., the smaller variance and smaller reciprocity in enmity networks) suggests possibly divergent perceptional biases. For example, since the paradox strength is proportional to the degree variance, it could be lower for enmity networks; however, we observe similar paradox patterns in both enmity networks and in a mixed world of friendship and antagonistic ties.

There is also a fundamental connection between the generalized friendship paradox (involving non-topological features) and the enmity paradox. If we treat the number of enemies a person has as a non-topological attribute for the positive network, then the enmity paradox follows from the generalized friendship paradox. Looking at the correlation between positive and negative degrees reveals these observations (Appendix, Fig. S10). As a result of the positive correlation between positive and negative degrees in most of the village networks ($\textrm{Pearson’s correlation}=0.18$, $p<10^{-16}$; in other words, the empirical reality that people who have many friends also tend to have more enemies), we are able to actually answer the question of why the enmity paradox exists in the first place through the lens of the generalized friendship paradox.

These paradoxes have further implications. Our understanding of social norms and of our social standing is influenced by our perceptions regarding those around us. For instance, the friendship paradox can help explain systematic biases in social perceptions such as regarding the prevalence of binge drinking and risky behaviors [16, 26]. Furthermore, the friendship paradoxes can explain why a given behavior in a society can be amplified. This can occur in two interwoven phases, where popular individuals act more intensely for activities associated with strategic complementarities, and those who are prone to certain behaviors interact more with other people who are also involved in that behavior, amplifying the effects of this behavior [16]. As a result, perceptions of behavior increase and this could contribute to an increase in the behavior along the lines of the perception. Misperceptions about the habits of one’s enemies could act similarly.

Such inter-personal influence, even if biased, can in turn be exploited to foster cascades, as field experiments have shown [10, 11, 13]. Indeed, as a result of phenomena like the friendship and enmity paradoxes, we could further perfect network targeting algorithms that exploit the friendship paradox; and taking into account a person’s enemies could make network targeting even more effective.

Biases associated with our antagonistic ties could be consequential in still another way. For instance, the friendship paradox may intensify homophilous patterns in network formation due to the misperception [16, 27]. Because of these paradoxes and the misperceptions they can give rise to, people might form a miscalbirated perception of their own attributes and thus seek out connections with people different than they might otherwise truly wish or “deserve.” The enmity paradox might similarly change our perceptions of reality and may function as a deterrent force in the formation of network ties between a person and the social connections around a person’s antagonists. When people judge whom to either befriend or avoid, they may be biased in their perceptions.