When is a network “decentralized enough?”

Posted 08/08/2023

I’ve submitted a new paper! Here’s the not-peer-reviewed pre-print. This post will discuss my work for non-network-scientist audiences.

There is broad disillusionment regarding the influence major tech companies have over our online interactions. Social media is largely governed by Meta (Facebook, Instagram, Whatsapp), Google (YouTube), and Twitter. In specific sub-communities, like open source software development, a single company like GitHub (owned by Microsoft) may have near-monopolistic control over online human collaboration. These companies define both the technology we use to communicate, and thereby the actions we can take to interact with one another, and the administrative policies regarding what actions and content are permissible on each platform.

In addition to debates over civic responsibility and regulation of online platforms, pushback to the centralized influence of these companies has taken two practical forms:

  1. Alt-Tech. Communities that are excluded from mainstream platforms, often right-wing hate and conspiracy groups, have built an ecosystem of alternative platforms that mirrors their mainstream counterparts, but with administrations more supportive of their political objectives. These include Voat and the .Win-network (now defunct Reddit-clones), BitChute and Parler (YouTube-clones), Gab (Twitter-clone), and many others.

  2. The Decentralized Web. Developers concerned about centralized control of content have built a number of decentralized platforms that aim to limit the control a single entity can have over human communication. These efforts include Mastodon, a Twitter alternative consisting of federated Twitter-like subcommunities, and ad-hoc communities like a loose network of self-hosted git servers. The decentralized web also encompasses much older decentralized networks like Usenet and email, and bears similarity to the motivations behind some Web3 technologies.

It is this second category, of ostensibly self-governed online communities, that interests me. Building a community-run platform is a laudable goal, but does the implementation of Mastodon and similar platforms fall short of those aspirations? How do we measure how ‘decentralized’ a platform is, or inversely, how much influence an oligarchy has over a platform?

The Community-Size Argument

One common approach to measuring social influence is to examine population size. The largest three Mastodon instances host over half of the entire Mastodon population. Therefore, the administrators of those three instances have disproportionate influence over permitted speech and users on Mastodon as a whole. Users who disagree with their decisions are free to make their own Mastodon instances, but if the operators of the big three instances refuse to connect to yours then half the Mastodon population will never see your posts.

A size disparity in community population is inevitable without intervention. Social networks follow rich-get-richer dynamics: new users are likely to join an existing vibrant community rather than a fledgling one, increasing its population and making it more appealing to future users. This is fundamentally a social pressure, but it is even further amplified by search engines, which are more likely to return links to larger and more active sites, funneling potential users towards the largest communities.

But is size disparity necessarily a failure of decentralization? Proponents of Mastodon have emphasized the importance of small communities that fit the needs of their members, and the Mastodon developers have stated that most Mastodon instances are small, topic-specific communities, with their mastodon.social as a notable exception. If smaller communities operate comfortably under the shadow of larger ones, perhaps this is a healthy example of decentralized governance.

Before exploring alternative methods for measuring social centralization, let’s compare a few of these decentralized and alt-tech platforms using the lens of community size. Below is a plot of sub-community population sizes for five platforms.

The y-axis represents the population of each community as a fraction of the largest community’s size. In other words, the largest community on each platform has a size of “1”, while a community with a tenth as many users has a size of “0.1”. The x-axis is what fraction of communities have at least that large a population. This allows us to quickly show that about 2% of Mastodon instances are least 1% the size of the largest instance, or alternatively, 98% of Mastodon instances have fewer than 1% as many users as the largest instance.

This puts Mastodon in similar territory as two centralized platforms, BitChute and Voat. Specifically, the number of commenters on BitChute channels follows a similar distribution to Mastodon instance sizes, while the distribution of Voat “subverse” (analogous to “subreddits”) populations is even more skewed.

By contrast, the number of users on self-hosted Git servers (the Penumbra of Open-Source), and unique authors on Polish Usenet newsgroups, is far more equitable: around a third of git servers have at least 1% as many users as the largest, while the majority of newsgroups are within 1% of the largest.

Inter-Community Influence

If smaller communities exist largely independently of larger ones, then the actions of administrators on those large communities does not matter to the small community, and even in the face of a large population disparity a platform can be effectively decentralized. How can we measure this notion of “independence” in a platform-agnostic way such that we can compare across platforms?

Each of the five platforms examined above has some notion of cross-community activity. On Mastodon, users can follow other users on both their own instance and external instances. On the other four platforms, users can directly participate in multiple communities, by contributing to open source projects on multiple servers (Penumbra), or commenting on multiple channels (BitChute), subverses (Voat), or newsgroups (Usenet).

In network science terminology, we can create a bipartite graph, or a graph with two types of vertices: one for communities, and one for users. Edges between users and communities indicate that a user interacts with that community. For example, here’s a diagram of Mastodon relationships, where an edge of ‘3’ indicates that a user follows three accounts on a particular instance:

This allows us to simulate the disruption caused by removing an instance: if mastodon.social went offline tomorrow, how many follow relationships from users on kolektiva.social and scholar.social would be disrupted? More globally, what percentage of all follow relationships by remaining users have just been pruned? If the disruption percentage is high, then lots of information flowed from the larger community to the smaller communities. Conversely, if the disruption percentage is low, then users of the smaller communities are largely unaffected.

Here is just such a plot, simulating removing the largest community from each platform, then the two largest, three largest, etcetera:

From this perspective on inter-community relationships, each platform looks a little different. Removing the largest three Mastodon instances has a severe effect on the remaining population, but removing further communities has a rapidly diminished effect. Removing Usenet newsgroups and BitChute channels has a similar pattern, but less pronounced.

Voat and the Penumbra require additional explanation. Voat, like Reddit, allowed users to subscribe to “subverses” to see posts from those communities on the front page of the site. New users were subscribed to a set of 27 subverses by default. While the two largest subverses by population (QRV and 8chan) were topic-specific and non-default, the third largest subverse, news, was a default subverse with broad appeal and high overlap with all other communities. Therefore, removing the largest two communities would have had little impact on users uninvolved in QAnon discussions, but removing news would impact almost every user on the site and cuts nearly 10% of interactions site-wide.

The Penumbra consists of independently operated git servers, only implicitly affiliated in that some developers contributed to projects hosted on multiple servers. Since servers are largely insular, most developers only contribute to projects on one, and so those developers are removed entirely along with the git server. If a user contributed to projects hosted on two servers then disruption will increase when the first server is removed, but will decrease when the second server is removed, and the developer along with it. This is shown as spiky oscillations, where one popular git server is removed and drives up disruption, before another overlapping git server is removed and severs the other side of those collaborations.

Sometimes you may be uninterested in the impact of removing the largest 2, 3, or 10 instances, and want a simple summary statistic for whether one platform is “more centralized” than another. One way to approximate this is to calculate the area under the curve for each of the above curves:

This scores Mastodon as the most centralized, because removing its largest instances has such a large effect on its peers. By contrast, while the Voat curve is visually striking, it’s such a sharp increase because removing the largest two communities doesn’t have a large impact on the population.

Situating Within Network Science

“Centralization” is an ill-defined term, and network scientists have a range of ways of measuring centralization for different scenarios. These metrics fall into three broad categories:

Scale Description Examples
Vertex Measures how central a role a single node plays in the network Betweenness centrality, Eigenvector centrality
Cluster Measures aspects of a particular group of vertices Assortativity / Homophily, Modularity, Insularity / Border index
Graph A summary attribute of an entire graph Diameter, Density, Cheeger numer

These metrics can capture aspects of centrality like “this vertex is an important bridge connecting two regions of a graph” or “this vertex is an important hub because many shortest paths between vertices pass through it.” They can measure how tight a bottleneck a graph contains (or, phrased another way, how well a graph can be partitioned in two), they can measure how much more likely similar vertices are to connect with one another, or how skewed the degree distribution of a graph is.

However, these metrics are mostly intended for fully connected unipartite graphs, and do not always have clear parallels in disconnected or bipartite graphs. Consider the following examples:

Many would intuitively agree that the left-most graph is central: one community in the center is larger than the rest, and serves as a bridge connecting several other communities together. By contrast, the middle graph is decentralized, because while the communities aren’t all the same size, none are dramatically larger than one another, and none serve a critical structural role as a hub or bridge.

The graph on the right is harder to describe. One community is much larger than its peers, but the remaining graph is identical to the decentralized example. By degree distribution, the graph would appear to be centralized. If we add a single edge connecting the giant community to any user in the main graph, then the giant community’s betweenness centrality score would skyrocket because of its prominent role in so many shortest-paths between users. However, it would still be inappropriate to say that the largest community plays a pivotal role in the activity of the users in the rest of the graph - it’s hardly connected at all!

My disruption metric is a cluster-level or mesoscale measurement for bipartite graphs that measures the influence of each community on its peers, although you can calculate the area under the disruption curve to make a graph-scale summary statistic. Using this approach, the centralized community is decidedly centralized, and the decentralized and ambiguous graphs are decidedly not.


Community size disparity is natural. Some communities will have broader appeal, and benefit from more from rich-get-richer effects than their smaller, more focused peers. Therefore, even a thriving decentralized platform may have a highly skewed population distribution. To measure the influence of oligarchies on a platform, we need a more nuanced view of interconnection and information flow between communities.

I have introduced a ‘disruption’ metric that accounts for both the size of a community and its structural role in the rest of the graph, measuring its potential influence on its peers. While the disruption metric illustrates how population distributions can be deceptive, it is only a preliminary measurement. Follows across communities and co-participation in communities are a rough proxy for information flow, or a network of potential information flow. A more precise metric for observed information flow might measure the number of messages that are boosted (“retweeted”) from one Mastodon instance to another, or might measure how frequently a new discussion topic, term, or URL appears first in one community, and later appears in a “downstream” community.

Does population size correlate with these measurements of information flow and influence? Are some smaller communities more influential than their size would suggest? How much does the graph structure of potential information flow predict ‘social decentralization’ in practice? There are many more questions to explore in this domain - but this is a start!