Natural ecosystems regularly have to endure disruptive external forces. However, when given time to recover, the ecosystems often find a new stable equilibrium. The network of interactions within an ecosystem is constructed in such a way that it facilitates stability against external influences, unlike random networks. This article will provide an extensive review of the academic ecology research to gain insight in why ecosystem networks are so stable while random networks are not. In addition, network biomimicry will be applied to take lessons from these ecosystem networks and apply them to stabilize human constructed networks, with a focus on the human interaction networks in a Decentralized Autonomous Organization (DAO).

First, the concept of network biomimicry will be introduced. Afterwards, different interaction types in ecosystems will be described, as well as how they can be represented as networks. Next, different measures for network stability will be presented. That sets us up for an overview of the history of research into the complexity-stability relationship of an ecosystem, addressing various metrics related to network complexity. Initially, different interaction types in an ecosystem will be described separately and afterwards they will all be brought together in a multi-layered network.

Throughout the article, lessons taken from ecosystem networks will be applied to DAOs. The presented information is there to inspire DAO contributors and founders and is in no way intended to be financial and/or legal advice.

Network biomimicry

In my previous article, I presented the concept of network biomimicry. This means taking inspiration from networks that are present in nature when constructing our human made networks. Human made networks exist in various forms, like logistic networks or social interaction networks. With the ongoing technological advancements that support an increase in globalization, these human made networks become increasingly more complex and unlike anything we have encountered before. Taking inspiration from complex networks in nature could help guide us in creating our own complex human made networks.

Complex biological networks exist at various spatiotemporal scales, like the network of biomolecules forming a cell; the network of neurons and glial cells that together constitute the brain; or the network of interacting species that populate an ecosystem. Despite differences in their size and the timescales at which they operate, these networks have a lot in common.

The introductory article about network biomimicry, presented examples of commonalities that can be found in these different biological networks. Furthermore, it described how these commonalities could be applied to strengthen a newly emerging type of social interaction networks, the DAO.

Some of the examples related to the overall network architecture of the DAO:

• Making it clustered
• Varying the extent of plasticity between clusters
• Incorporating feedback loops

Other examples related to the optimal way in which individuals can contribute to the DAO:

• Being their authentic selves
• Connecting and collaborating with others
• Balancing strong and weak interactions

An ecosystem perspective

In this article, we’ll dive deeper into the relationship between the size and architecture of a network and its stability. Ecosystem networks will be the main focus due to the fact that data regarding the relationship between the complexity and stability of an ecosystem is relatively easily available. Thus, this has been a topic of academic research for decades.[i] [ii] [iii] Both field measurements as well as computational modeling studies will be considered in this review of the academic literature.

When researching the content for this article, the excellent review paper “Complexity and stability of ecological networks: a review of the theory” from Landi et al. has been a big source of inspiration. I want to thank the authors for putting together such an elaborate but concise overview of the history of theoretical ecology. For the scope of this article, I chose to leave out the mathematical aspects but I want to guide anyone who is interested in learning more about this topic to their paper.[iv]

Ecosystems as a network

Ecosystems consist of various species that interact with each other. These interactions can be of different forms.

• Antagonistic interactions: one species is harmed while the other benefits. For example, the predator prey interaction between a lion and a gazelle. The lion hunts on the gazelle in order to be able to proliferate.
• Mutualistic interactions: both species benefit from each other. Like the interaction between a flower and a pollinator. The flowers provide food for the pollinators which in turn help in the reproductive process of the flower.
• Competitive interactions: both species negatively affect each other. For instance, two different sea mammals that compete for the same limited food sources.

These different interaction types can be presented in the form of a network. Here the species are points in the network, for example the lion or the gazelle. Interactions between the species are presented as connections between the points. For instance, the predator-prey interaction between the lion and the gazelle. The points in the network will be referred to as nodes. The connections in the network will be referred to as edges. The strength of an edge indicates the extent of the impact one species has on the other.

The image below shows examples for the different interaction types. These examples only consist of one pair of species/nodes. Actual ecosystems can be seen as networks of these types of interactions.

Stability measures

The stability of an ecosystem network indicates how it responds to perturbations. The network is considered stable if it returns to an equilibrium, where the population sizes of all species are maintained at constant abundances. If instead, the population sizes continue to evolve further and further away from an equilibrium, the network is considered unstable.[v] [vi] The Resilience of an ecosystem network reflects the time it takes to return to an equilibrium after a perturbation.[vii] [viii]

Stability in response to removal

A perturbation could be a species (node) that gets removed from the network. From a DAO perspective, this could be a contributor leaving the DAO. Other species/DAO members might rely heavily on the presence of the removed node and could in turn also go extinct/leave the DAO. In ecology this is referred to as an extinction cascade. The persistence of a network reflects the number of species that remain after the new equilibrium is reached.[ix] [x]

With computer modeling this can be quantified, by removing species from the modeled ecosystem and measuring the loss of additional species resulting from the removal. The robustness of a system is a measure for the ability of the system to resist extinction cascades.[xi]

Stability in response to addition

Another type of perturbation is invasion of a new species. From a DAO perspective, this could be a large number of inexperienced members joining a DAO. The resistance to invasion describes the resistance of an ecosystem in response to the invasion of a new species.[ii]

The history of the ecological complexity-stability debate

Now that we have introduced some of the main concepts related to the stability of a network, we are all set to explore the scientific debate regarding the influence of the complexity of the network on its stability.

In the early days of ecology, research was solely conducted based on observational studies in healthy and perturbed ecosystems. It was observed that more complex ecosystems also tended to be more stable. Diverse ecosystems where thought to be more resilient against invasions of new species.[i] And a higher variety in predators and parasites was proposed keep the ecosystem at an equilibrium by preventing populations of other species to undergo explosive growth.[ii] Furthermore, it was claimed that stability was easier to achieve in ecosystems that were more numerous in the ways in which its species interacted.[iii]

Later on, mathematical simulations were included in the scientific toolbox of the ecologist. Early studies that modeled ecosystems resulted in opposing findings regarding the relation between the complexity and stability of a network. In these works, matrices were created where each row entry (i) described the impact of a specie on the growth of the species on the columns (j). With all the species in the ecosystem represented on both the rows and the columns, every possible interaction (ij) between species is described in the matrix.

Assuming the ecosystem to rest at an equilibrium, small perturbations were made after which it was studied if the system would return to a new equilibrium or not. This analysis showed that an increase in the number of species and/or the number of connections between species increases the chances of the network to exhibit unstable behaviors.[v] [vi]

An important remark that should be made about these modeling studies is that they were performed using randomly generated interaction networks. Interestingly, modeled random networks showed opposite results as that were observed in ecosystems. The random networks became less stable with increasing complexity while complexity made ecosystems more stable. These findings indicate that the ecosystem networks have certain features that distinguish them from random networks. And these features cause the ecosystem networks to become more stable with increasing complexity instead of less.

Learning what these features are can be insightful when constructing other types of stable complex networks, like the interaction networks between members of a DAO. In the next section, an overview is given of the relation between different network features and different network stability measures for antagonistic, mutualistic and competitive networks. Afterwards, the combination of these different networks into one multilayer network and its effects on the stability of the network will be discussed. The lessons learned from these observations will be related to DAOs in each section.

Network complexity

Network size

The complexity of a network can be described in various ways. A straightforward metric is the size of the network, indicated by the total number of nodes. Within the ecosystem perspective, this would reflect the total number of different species in the network.[iii] For a DAO, this could be the total number of active contributors.

When only considering feasible network models for the stability modeling analysis, larger antagonistic networks still resulted in a reduced stability.[ix] Opposing results were observed in mutualistic networks, where a large network size promotes the resilience and persistence of the network.[vii] [viii] Similarly, the number of competing species in an ecosystem has been reported to be positively related to the stability of the ecosystem.[xiii] These observations were later confirmed through theoretical modeling of competitive networks.[xiv]

These results indicate that increasing the complexity of predation has a destabilizing impact whereas increasing collaboration and mutual competition are stabilizing. This is already present in many DAOs where DAO members contribute their unique skills in a collaborative effort, working towards the overarching goals of the DAO (mutualistic). For instance, in the form of guilds consisting of DAO members that together achieve specific goals for the DAO. At the same time, DAO members can compete with each other in coming up with the best solutions and implementations (competitive). For example, by filing competing proposals in attempts to obtain a bounty, promoting high quality proposals. However, scenarios where certain DAO members gain as a direct result of bringing down other DAO members (antagonistic) are hardly encountered.

Connectance

Another related metric is the connectance, which reflects the proportion of realized edges in the network. For an ecosystem/DAO, this would indicate the extent to which the different species/members affect each other. Every edge describes the interaction between two species/members and so a higher proportion of realized edges indicates that there is more interaction.[v]

Increasing the connectance between species in an antagonistic network also increases the network stability. And this effect becomes stronger when the connectance is higher.[xv] More specifically, connectance between weakly and strongly self-regulated elements of the network is especially important for stability in antagonistic networks.[xvi] [xvii]

It was also shown that highly connected mutualistic networks increased the resilience of the network[vi] [vii] as perturbations can propagate less easily through these networks.[xviii] Connectance is positively correlated with persistence in these networks too.[xix] On a contrary note, extinction cascades are also more likely to happen in highly connected mutualistic networks.[xi]

Interestingly, a 12 year longitudinal study of a mutualistic ecosystem network showed that the connectance remained stable over this time period, despite significant turnover of species and interactions. This further emphasizes the relevance of the connectance in a network.[xx] Finally, a higher connectance increases the stability of competitive networks, regardless of whether these connections are weighted or not.[xxi]

In all cases, a higher connectance leads to a more stable network. This highlights the importance of abundant information flows among the members in a DAO. However, caution should be taken when it comes to the connectance of mutualistic interactions. Where intermediate extends of collaboration are important, having the operations of a DAO be too dependent on the contributions of a large number of individuals can destabilize the DAO when one or several of these individuals decides to leave the DAO.

Node degree distribution

More detailed measures of complexity look at the differences between individual nodes in the network. For instance, the variation in the number of edges that a node has with other nodes in the network.[xxii] The number of edges that a node has with other nodes is called its degree.

A distribution can be made of the degree values of all the nodes in the network. For an ecosystem, this distribution could show whether all species interact with about the same number of other species or whether some species in the ecosystem interact with a lot of other species (generalists) while others only interact with a selected few (specialists). Similarly for a DAO, the node degree distribution can give you an idea whether everyone only interacts with part of the other members of the DAO or whether some of the DAO members might be connected with nearly all other members.

The distributions of node degree in ecological networks have been shown to differ from the distributions you would expect for randomly generated networks.[xxiii] Skewed node degree distributions, especially exponential degree distributions, make antagonistic networks more stable in response to removals.[xi] [xxiv] [xxv] Broad degree distributions, with large differences between the number of connections that each node has, also tend to stabilize antagonistic networks.[xxvi]

Most mutualistic networks were found to have node degree distributions that fit a truncated power-law. This distribution suggests the prevalence of specialists with only a few connections while indicating the rarity of super generalists with many connections.[xxvii] On the other hand, it was also shown that a heterogeneous node degree distribution (like power-law or exponential distributions) negatively affecting the stability of mutualistic networks.[xxviii]

Generally speaking, having variation between different DAO members when it comes to the number of other DAO members that they interact with is important. High degree generalists that interact with a large number of other DAO members are key. Take the DAO founders or council members as example. Meanwhile low degree specialists that interact with a smaller number of other DAO members are important too. For instance, DAO members with a specific expertise applied to a specific task. Being a low degree node in a number of different DAO networks allows these members to be experts in what they do.

Ecosystem analyses imply that super generalists, which are directly connected with nearly all DAO members, should be avoided. This would apply especially to larger more matured DAOs that have already grown into their fully decentralized form. Finally, when it comes to collaboration, having either too little collaborative interactions or too many collaborative obligations as a DAO member can negatively impact overall collaboration within the DAO.

Edge strength distribution

A different measure, but related to the node degree distribution, is the edge strength distribution. This distribution visualizes the values of the edges in the network.[xxii] These values represent the strength of existing interactions between all pairs of nodes.

This distribution can be studied to see whether the magnitude of the influence of one species on another is similar among all interactions or whether certain interactions have a larger impact than others. This metric can also tell you more about the depth of interactions between different DAO members. Are some of these interactions strong, indicating a close collaboration between these two DAO members? Or are the interactions mainly weak indicating more shallow interactions between members?

A skewed distribution of interaction strengths with many weaker interactions and a few strong ones have been observed in many antagonistic networks.[xxix] [xxx] [xxxi] [xxxii] The weak connections in these networks were found to make them more stable[viii] [x] [xvii] (although opposing results have been reported as well[xxxiii] [xxxiv]).

Similar skewed interaction strength distributions were also observed in mutualistic networks[xxxv] and suggested to contribute to the stability of these networks.[xxxvi] Although there is likely a limit to the optimal skewedness as it was later shown that heterogeneity of the interaction strength distribution can have a negative impact on the network stability.[xxviii] Finally, a skewed interaction strength distribution with many weaker interactions was also shown to stabilize competitive networks and enhance their robustness.[xxxvii]

Applying these findings to DAOs indicates the importance of having both a framework of strong edges among DAO members that regularly interact and work together, as well as having a larger body of weaker edges among DAO members. These weaker edges could represent DAO members that have more variation in the members that they interact with. Such a balance can allow the DAO to simultaneously be resilient and agile. Similar distributions have been observed in other biological systems too, for instance when studying interaction strengths between neurons.[xxxviii]

For both the interaction strength distribution and the node degree distribution, it is important to keep in consideration that these distributions cover a spectrum. And all parts of the spectrum are important. Solely implementing the boundaries of the spectrum by having only very strong edges / high degree nodes and very weak edges / low degree nodes can be destabilizing.

Clustering and small-worldness

Finally, there are also complexity metrics that reflect the architecture of the network. One of the most important architecture metrics is the level of clustering. Clustering reflects the extent to which the network is divided into separate clusters consisting of nodes that are highly interconnected with each other and to a lesser extend connected with other nodes.[xxxix] This measure indicates the extent to which the ecosystem or DAO is compartmentalized into different groups.

The stability of an antagonistic network is enhanced in clustered architectures.[viii] [xl] [xli] Furthermore, persistence also increases with clustering.[xlii] Although it was later shown that these observations only hold under specific circumstances.[xliii]

Observational studies also reported a clustered structure in various types of mutualistic networks, with the level of clustering increasing with network size.[xxxix] [xliv] Furthermore, mutualistic networks tend to be highly nested[xlv] which was shown to contribute to the resilience[vii] [viii], persistence[xlvi] and robustness[xlvii] of the network (although opposing claims have been made as well[xix] [xxxiii]). Nestedness is a special type of clustering: the nodes in a nested cluster only interact with part of the other nodes in the network whereas nodes outside of the cluster can interact with all the other nodes. There is likely a limit to nestedness as it was shown that extreme nestedness makes the mutualistic network more susceptible to extinction cascades.[xlviii]

Over-clustering can increase the average shortest path length in a network. This measure indicates the average number of edges it takes to move from any node in the network to any other node. From an ecosystem perspective, it can give an indication to what extent a change to one species in the ecosystem impacts other species. Similarly, it can tell you how easily ideas can travel from one DAO member to another. The balance between the extent of clustering and the average shortest path length has a metric of it’s own called the small-worldness of a network. Small-worldness is observed in various biological networks like ecosystems[xlix] but also cells[l] and the brain[li].

Clustering can positively contribute to a DAO as it allows for effective exchange of information on a local scale, for instance within a guild. However, for the DAO as a whole it is important that there is efficient information exchange between the clusters. Otherwise, the DAO becomes disconnected and easily falls out of balance. Therefore, a clustered small-world structure can help a DAO find an optimal balance between local information processing and global information processing. This can, for example, help prevent the DAO from being slowed down too much by decision making processes while still keeping the decision-making process decentralized.

Multilayer networks

So far, antagonistic, mutualistic and competitive networks have been discussed separately. However, this is not a true representation of an ecosystem where all these interactions occur simultaneously. By perceiving these different types of interaction networks as individual layers in a larger multilayer network, a more accurate representation of the ecosystem can be made.[lii]

Comparing the robustness to perturbations between monolayer networks and multilayer networks shows that extinctions occur more slowly in multilayer networks.

For example, consider scenario 1: a mono layer antagonistic network between plants and plant parasites. Gradually removing plants from the network leads to the extinction of parasites.

Now consider scenario 2: a multilayer network consisting of a mutualistic layer with plants and pollinators and an antagonistic layer with plants and parasites. Gradually removing pollinators from the network leads to the extinction of plants which leads to the extinction of parasites. In this second scenario, the extinctions of parasites occur more slowly (see a in figure below). Interestingly, the extinction rate of the plants remained more or less the same in the 2 scenarios (see b in figure below).

Finally, consider scenario 3: the multilayer network is the same as in scenario 2 but now both plants and pollinators are gradually removed from the multilayer network. The proportion of surviving parasites still remained higher than in scenario 1 with a single layer network consisting of only plants and parasites.[xii]

Other research showed that including antagonistic interactions in multilayer networks actually results in an increased stability when the network size and connectance increase.[liii] This observation opposed the earlier described work studying the stability of a fully antagonistic network in response increasing the network size.

These examples highlight the importance of having different types of interactions present in a network. Creating space for both collaboration and competition within a DAO could induce overall stability.

In addition, including some kind of antagonistic interactions could further help stabilize the DAO. However, in the DAO context, assigning this role to individual DAO members could potentially be dangerous as it can induce a hostile environment. Instead, these interactions could be included into the DAO protocol. For instance, by defining time limits that a certain contributor can hold a certain role or by describing ways in which guilds or groups can be pruned away from the DAO when they become outdated or irrelevant.

Conclusion

A review of the literature on the relation between network complexity and stability has shown that in modeled random networks, the network stability decreases with increasing complexity. However, in actual ecosystems, the opposite is observed. In this article, we went over some features of these ecosystem networks that distinguish them from random networks. These features could help making human constructed networks like DAOs more stable while they grow in size and complexity. Some important lessons learned from ecosystems are:

• Increasing the number of nodes in a mutualistic network or a competitive network makes it more stable while increasing the number of nodes in a fully antagonistic network makes it less stable.
• A higher connectance within the network enhances stability.
• Nodes interacting with a large number of other nodes (high degree) as well as nodes interacting with a low number of other nodes (low degree) both have their own important role in a stable network.
• A balance between some strong edges and a larger body of medium and weak edges further stabilizes the network.
• Clustering also stabilizes a network, although only when present in moderation. Finding the optimal balance between clustering and short path lengths, described by the small-worldness metric, enhances information flow within the network.
• Combining different interaction types into a multilayer network further enhances stability. In this case, antagonistic interactions can even contribute to more stability, unlike in purely antagonistic networks.

This is the second article in a series about network biomimicry and DAOs. If this perspective on DAOs and other human made networks is interesting to you, feel free to reach out. You can find me on LinkedIn (Tjitse van der Molen).

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