About dyads and triads

Usually when we think about social relationships, we think about dyads. We think about the attributes of a relationship between one individual and the other. There could be many ways to describe a social relationship, including its strength, symmetry, predictability, etc. A detailed review of these properties can be found in this interesting paper by Joan Silk et al. However, sometimes social decisions may not be dependent only on the properties of the two individuals involved in a dyad, but rather on constraints imposed by the social structure. Here comes to effect the relationships they have with other individuals. One such constraint is described by the theory of structural balance, formalizing the saying "the friend of my friend is my friend". The idea is that a dyad may be influenced by a 3rd party. If I am a friend of John, and he is a friend of Alex, their friendship affects my decision about becoming a friend of Alex. If I don't like Alex, that will create a tension between me and my friend John. For example, if John is having a party, I would not like him to invite Alex. This situation is called an unbalanced triad. In contrast, if both John and me do not favor Alex, there is no social tension in the triad, since John and I have the same opinions. There is no incentive for any of us to make a change.

The theory of structural balance was developed by Austrian psychologist Heider back in 1946. It was later formally applied to social networks by Cartwright & Harary. It was found to be relevant for human societies, and also for international relationships. One study suggested that the lack of balance was the trigger behind World War I.

I tested if structural balance may also be relevant for animal societies. I thought that similar social tensions as in humans may affect animals. In a recent paper, we found that rock hyraxes do seem to care about it. We used data from our long-term study, and compared the social network to random networks retaining some properties of the observed ones. We found that balanced triads were more common than expected by chance, while unbalanced triads were less common than expected. Interestingly, we found what is called "weak" structural balance. That means that triads in which all three individuals were not in positive relationship were more common than expected. In contrast, strong structural balance predicts that such triads should be unbalanced, and two individuals are expected to form an alliance against the other. I think that our finding is more relevant to animal societies, in which there are usually more than two groups in a population. Thus a balanced animal network is usually composed of groups, in which in general individuals favor each other, but are not associated with individuals from other groups.

To test if structural balance is not merely a consequence of another process, we followed each triad across years. First, we found a marginally significant effect of triad type on survival. Hyraxes in unbalanced triads had a lower chance of surviving to the following year. Then we found that balanced triads usually stayed in the same configuration, while unbalanced triads mostly changed to balanced ones. That suggests that hyraxes "care" about the triads, and actively change the unbalanced ones.

The implication of a balanced network is the clustering of individuals into groups of cooperators. Thus, structural balance could be a mechanism involved in the widely observed phenomena of living in groups.

Paper: Socialism rules in the rock hyrax

Recent studies of humans show that more socialist countries, in which the assets are more evenly spread among the citizens, are better places to live in. Theses countries demonstrate higher longevity, less crime, less health problems and so on. In contrast, in these countries there are more suicides and more alcoholism. The explanation is that in such countries people direct violence towards themselves, and not towards other people. In countries with large gaps between the rich and the poor, all suffer: it's not only the poor who are victims of violence and health problems.

In my latest paper, Adi Barocas and me looked for the effects of social network structure on longevity in the rock hyrax. Our long term study allows us to follow individuals throughout their lives and determine their time of death. We found that groups vary in their equality of social associations: some groups are more "egalitarian", while in others there are highly connected individuals alongside peripheral ones. Our main finding is that hyraxes living in more egalitarian groups lived longer, which means that such groups present some advantages for their members. More technically, if the variance in strength centrality was high the hyrax had lower chances to survive. Interestingly, we found no such effect in the individual level. In other words, in the socially skewed groups, it's not necessarily the more central individuals that survived longer. Social inequality affected all members. Therefore, if we are looking for the mechanism that generates this outcome, I believe that it is not related to predation. If predation was the issue, we would expect to find that less connected individuals died sooner, but that was not found. Instead, I think that less "socialist" groups lead to higher stress, that may affect all members.

Rock hyraxes in Ein Gedi. Photo: Giora Ilany

There comes the question of why some groups cannot maintain social equality. I think it may be the result of a few dominant individuals, who are usually aggressive and increase the stress in the group, but this stress affects them too. As we continue to follow this population, we may be able to find the reasons for social inequality in some groups. It is worth to mention that the same group may change during the years, becoming less or more equal, and that is probably a result of its composition of members.

The higher groups have lower variance in centrality than the lower groups. Node size is proportional to centrality. 

In other findings, we show that group members survived better than solitary males, again demonstrating the advantages of sociality. We also found that individuals in smaller groups survived better than those living in larger groups. This shows that sociality has its limits, and groups wich are too large are not advantageous for their members. Group size was not related to variance in centrality.

Overall, I think that our results add to the growing literature showing the effect of sociality on fitness, as previously found in baboons and dolphins. In addition, this study is one of the first to examine the effects of social structure in the group level.

Paper: Who Spreads Parasites in Lizards?

Little is known about how parasites are spread through in host populations. Spread patterns should depend on host contact patterns. Social network analysis provides great tools for modeling parasite and pathogen transmission in host populations. Theoretically, highly connected individuals are in greater risk of being infected.

In a recent paper in the Journal of Animal Ecology, Fenner et al. describe the infection patterns of a tick and a nematode in the pygmy bluetongue lizard (Tiliqua adelaidensis), a solitary lizard from South Australia. These lizards don't move much, staying in burrows most of the time.

A relative: the blotched bluetongue lizard

The authors used 3 plots representing sub-populations in the same area. In the only plot in which ticks were found on lizards, lizards that had more or closer neighbors, i.e. more connected, had more ticks. Stable resident hosts were more important than dispersers in influencing tick distribution.

For nematodes there was support for the role of dispersers - infected hosts were more connected to dispersers. This suggests there are different transmission pathways for different parasites, possibly due to differences in parasites survival times.

To summarize, this paper shows the strength of network analysis in exploring alternative hypotheses about the dynamics of infection patterns. It also exposes once again the dependence of the result on the way the network was defined - in this case the distances between burrows of lizards.

Paper: Network Analysis of Songbird Dialects

Today I am glad to present a paper by Yoktan et al. to which I have contributed the network analysis. This work describes the different dialects of songs of the orange-tufted sunbird in Israel. More than 100 years ago, sunbirds were found only in the southernmost parts of the Rift Valley in Israel, in the Arava and some oases to the north, such as Ein Gedi. The Zionist settlement in many small villages (kibbutz & moshav) along the Rift allowed the expansion of sunbirds as gardening in these settlements introduced many species of ornithophilous plants.

In this study we have recorded singing male sunbirds in many locations along the Rift and analyzed the spectrogram of their trill component. We found that each location had a slightly different dialect, and built a network of locations according to their dialects. We used data of the distance between each two dialects as the basic matrix. Since we had a full matrix, we had to set some threshold in order to remove some of the ties and get a meaningful network. We set the threshold to be the largest distance that still allowed all locations to be connected as one component, in order to be able to relate each location to the others.

The trills in different locations

The network of locations by song distances revealed locations that are "connected", i.e. their songs are relatively similar, while other locations were "disconnected" - their songs were quite different from each other. This network shows that there are "communities" of locations with similar songs (we determined the communities using the Girvan-Newman algorithm). Some of these communities consist of locations from the same geographical region, while others have a mix of locations from different regions. The locations in the Arava valley, the natural habitat of the sunbirds before the expansion, all belong to the same community. From there began the habitat expansion to the north. Three of the most isolated locations in the network indeed represent villages in the extreme north, suggesting these sunbirds do not interact much with more southern populations. Also interesting are the locations along the dead sea: sunbirds were present in Ein Gedi before the settlements, and their songs closely resemble the Arava songs. However, the songs of nearby Kaliya and Mitzpe Shalem, which were settled in the 1970s', are found in a different community, together with northern settlements, suggesting that these two locations were inhabited by sunbirds coming back from the north, and not directly from Ein Gedi.

The network of locations according to sunbird songs

Another interesting finding is that network centrality, which depicts how central were locations of singers in the network, was negatively correlated with genetic variability. This fact implies that the most central locations in terms of song dialects, which are Ein Gedi, Bet Zera and Sede Eliezer, host established populations of sunbirds which resist intruders. These locations are probably a source of dispersal to other places.

Overall, our results support the historical processes hypothesis of dialect formation, which predict song dialects of nearby locations which were occupied at the same time to be similar. This work illustrates the power of network analysis in describing not only social relations between animals, but also other types of relations. I believe it could be useful for many other types of analyses. 

Degree Centrality - Mini Review

While reviewing the fast accumulating literature about animal social networks might be an overwhelming task, I would like to try and focus some posts on some basic network measurements and what we have learned from them so far. Degree is the most basic measurement for an individual in a network. In a binary network it is the number of ties an individual has with others. In a weighted network it is the sum of all individual's tie weights, which means that two individuals may have the same number of connections, but one of them may have stronger connections than the other.

So what do we know about animals in terms of degree? Lusseau and Newman (2004) showed that in a bottlenose dolphins population in New Zealand there is no assortative mixing by degree, i.e. dolphins having high degree do not preferentially attach to other dolphins with high degree. This is opposed to findings in humans, where assortative mixing is common. The dolphin network suggests that preferential attachment is not strong in their network evolution, meaning that the network was not formed by the connection of new members to central members. An additional finding of this study is that the network is robust to the loss of high-degree members. In other words, there are many redundant paths in the network, allowing it to withstand removal of central figures.
In a newer paper Lusseau et al. (2006) report some degree homophily in a bottlenose dolphin population in Scotland. It is difficult to say if the difference in these results reflects a real difference in the social structure of different dolphin populations or is an artefact of some methodological inconsistencies.

Croft et al. (2005) did find assortative interactions in a study of multiple guppy populations and one three-spined stickleback population. The degree of individual fish was positively correlated with the average degree of their network neighbors. The authors suggest that could cause faster spreading of information, but also of pathogens, among populations. Although this study was performed on multiple populations, we'll need more data in order to state that assortative interactions are common in animals.

Guppies

In another study, on captive pigtailed macaques, Flack et al. (2006) observed "policing" behavior by some members of a group. They showed that in the presence of policing the average degree of grooming and playing networks was higher. Policing also affected assortative mixing, where less policing meant increased assortativity. This is the first study to show the effect of individual removals on the degree of other individuals in the network. However, since this study was done on captive animals it has limited power in describing wild social structures.

A different kind of study compared the networks of onagers and Grevy's zebras (Sundaresan et al. 2007), and found that onagers have higher degree than zebra, i.e. they associate with more partners. Interestingly, when testing only preferred associations (after statistically testing which associations are more common than expected by chance), such differences were absent. This shows that these two species do not differ in the amount of significant associations each individual has.

Wolf et al. (2007) studied social networks of the Galapagos sea lion. I have reviewed a recent work by them in an earlier post. They found that males had lower average degree than females. The authors suggest that since females are less aggressive than males and show more fidelity to a specific location they may be able to form more associations.

As can be seen, we have only began to scratch the surface of what we can learn from the degrees of individuals. The described papers currently do not lead to any conclusion regarding other species, or even other populations. I will end this post here and continue in part 2.

Paper: Hamilton's r vs. Ohtsuki's k

A recent paper by Wolf et al. in The American Naturalist looks at the relation between genetic relatedness and social structure from a new angle. The authors compare two parameters that should predict the level of cooperation in a society. Hamilton's rule of kin selection (1964) suggests that an individual A should cooperate with B if the benefit to B, relative to the cost to A, is more than the inverse of the genetic relatedness between them (b/c > 1/r), i.e. that you'll have a stronger will to help a closer relative. Studying (rather simple) model networks, Ohtsuki et al. (2006) suggested that cooperation is favorable when b/c > k, k being the average number of neighbors in the network, meaning you'll have a stronger will to help someone if the network is more sparse.

In a study of sea lions in the Galapagos, they compared r and k in different social levels: individual, clique, and community. They found a strong negative correlation between r and k in all social levels, meaning that k and r capture similar structural information in the population.
It is suggested that individuals may prefer associating with their relatives, although it's hard to understand how they "know" their relatedness to any other individual (I can think of some olfactory mechanisms that may facilitate this).

The  authors raise the concern that k may not reliably represent K, the "real" average for the population over longer timeframes (data was collected over a 3-month period). I also share this concern, especially since only one population was used. However, from my experience, if three months are enough for getting a "saturated" network in the sea lions, then more observations would contribute little to the analysis. One more concern considers their use of filtered binary networks, where a lot of information is lost, especially about weaker associations. This analysis seems like one that could gain more power from a weighted network, where the strength of each tie is taken into account.

Community Detection

Of all topics in general network theory, my favorite is community detection. For some reason I can't stop reading about new algorithms, analyzing them and trying them on my data.

So what is community detection? Every network can be subdivided into sub-units which are groups of closely connected nodes (or individuals). In animal populations, communities can be groups of animals, couples or even solitary individuals. As usual, the categorization is dependent on the definition of a connection in the network. In the past researchers were using their own intuition or very basic calculations in order to decide which individual belongs to each group. While in some species its simple and clear to make this decision, in others, like the hyrax, it's not trivial. Some hyraxes are clearly part of a group, while others are occasionally seen out of the group, or even with another group. In these cases, it's not easy to decide to what group they belong. This decision has many consequences, as any analysis at the group level, or comparison of individuals from different groups, will depend on the definition of group members.

Community detection algorithms try to solve this problem by taking into account the connections (edges) each node has with all other nodes. There are many algorithms out there, reviewed by Fortunato. I'm not going to exhaustively review them - I would like to mention 3 algorithms I found to do an accurate job, that are relatively easy to use. One thing that is important to the mathematicians who invent most of these algorithms is their complexity, or the time it will take to run. For animal researches this is usually not a real problem, as the numbers of individuals/relations are small. Therefore, I don't care if an algorithm is heavy on resources.

The first algorithm, and one of the most famous in the field, is called Girvan-Newman, and was published in PNAS in 2002. This algorithm computes the betweenness centrality of each edge, and then removes the edge with the highest betweenness, assuming that it connects communities and cannot be inside a community. In the next step the betweenness of all edges is recalculated, and again the edge with the highest value is removed. When the algorithm stops the result is a tree describing the levels of connectedness. Later, a stop mechanism was developed, based on a measure called modularity, to try and decide when should the algorithm stops if the "right" number of communities is unknown. Girvan-Newman can be calculated using UCINet. Although it performs very well, there is some critique on the use of modularity, and newer algorithms were developed.

The second algorithm is called CPM, or clique percolation method (published in Nature in 2005). This algorithm works in a different way, trying to connect more and more cliques of closely related nodes. In contrast to the Girvan-Newman algorithm, this one allows for community overlap, so that one node can be part of more than one community. It works by starting with cliques of a given size k, and then adding to them the members of other cliques that share k-1 nodes, until no more cliques can be added. One disadvantage is that you have to define k, or test which k gives the best results, which brings us back to the question of what is the best result. Another drawback is its inability to find communities of size 2, as cliques contain 3 or more members. There are versions of this algorithm that work on directed or weighted networks. A software named CFinder implements this algorithm.

A result of CPM

The last algorithm for today is called Link Communities and was published in Nature in 2010. This one also allows for community overlap. Its idea is that many nodes belong to more than one community (for example, a person might be part of his family, his workmates, his friends at the bar, etc), but the links between individuals usually represent only one kind of connection. The algorithm gets complex in finding the right links to join, and produces very accurate results. It has no resolution problem, meaning you can find any number of communities you wish, but calculates the best answer in terms of partition density. Rob Spencer from Scaled Innovation created a beautiful implementation, which shows the communities of each node, and the partition density function:

All of these algorithms performed well on my hyrax data, usually accurately identifying the "right" members of each group.
I am sure that the last word was not said yet, and new algorithms are about to be developed. It will be interesting to see if and when this field gets saturated, and what algorithms will "win" and become standard.