Jose Betancourt
@jbetancourt015.bsky.social
Physics PhD student at Yale. Econ-adjacent. I’m fascinated by networks, complexity and chaos.
All comments/suggestions are welcome! I’m happy to hear what people think about this.
Here is a link to the paper: arxiv.org/abs/2510.10997
Here is a link to the paper: arxiv.org/abs/2510.10997
The Strength of Local Structures in Decentralized Network Formation
I study dynamic network formation games in which agents assign arbitrary values to network structures. Any such game admits an equivalent representation in terms of the values agents assign to its sub-structures, linking local valuations to equilibrium behavior. The game is a potential game precisely when all participants in a structure value it equally, yielding a closed-form stationary distribution. When valuations are restricted to a finite set of repeated sub-structures, or motifs, the model exhibits phase transitions: small changes in motif values cause discontinuous shifts in network density.
arxiv.org
October 14, 2025 at 4:18 PM
All comments/suggestions are welcome! I’m happy to hear what people think about this.
Here is a link to the paper: arxiv.org/abs/2510.10997
Here is a link to the paper: arxiv.org/abs/2510.10997
This result extends to heterogeneous agents, where types might correspond to location, sectors, etc. The same force that captures these amplifying effects is still there, which suggests that it might be lurking even in more complex models.
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15/15
October 14, 2025 at 4:18 PM
This result extends to heterogeneous agents, where types might correspond to location, sectors, etc. The same force that captures these amplifying effects is still there, which suggests that it might be lurking even in more complex models.
15/15
15/15
The intuition here is that when you are forming a connection, in addition to getting the instantaneous value from it, you increase the probability that structures are formed in the future. When these spillovers are reinforced enough, we can get phase transitions.
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October 14, 2025 at 4:18 PM
The intuition here is that when you are forming a connection, in addition to getting the instantaneous value from it, you increase the probability that structures are formed in the future. When these spillovers are reinforced enough, we can get phase transitions.
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Perhaps the most fascinating implication of this result is that networks can discontinuously respond to changes in motif values. This is a phenomenon called “phase transitions”. This means that even a small change in parameters can populate or destroy the network!
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October 14, 2025 at 4:18 PM
Perhaps the most fascinating implication of this result is that networks can discontinuously respond to changes in motif values. This is a phenomenon called “phase transitions”. This means that even a small change in parameters can populate or destroy the network!
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I show that the motif model converges to a (directed) Erdös-Rényi model, where the density solves the optimization problem below.
It intuitively says that agents try to form structures that maximize their motif values, but it is costly to explore the vast space of networks.
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It intuitively says that agents try to form structures that maximize their motif values, but it is costly to explore the vast space of networks.
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October 14, 2025 at 4:18 PM
I show that the motif model converges to a (directed) Erdös-Rényi model, where the density solves the optimization problem below.
It intuitively says that agents try to form structures that maximize their motif values, but it is costly to explore the vast space of networks.
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It intuitively says that agents try to form structures that maximize their motif values, but it is costly to explore the vast space of networks.
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We can specify a finite set of motifs, and this fully determines the incentives of players to form the network. This is useful to study how the long-run behavior of the process depends on the incentives to form structures.
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October 14, 2025 at 4:18 PM
We can specify a finite set of motifs, and this fully determines the incentives of players to form the network. This is useful to study how the long-run behavior of the process depends on the incentives to form structures.
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It could also be the case that if someone reciprocates a link, both individuals receive some benefit.
These are examples of motifs: recurring structures whose value doesn’t depend on who participates, only on the structure of connections. The examples above look like this:
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These are examples of motifs: recurring structures whose value doesn’t depend on who participates, only on the structure of connections. The examples above look like this:
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October 14, 2025 at 4:18 PM
It could also be the case that if someone reciprocates a link, both individuals receive some benefit.
These are examples of motifs: recurring structures whose value doesn’t depend on who participates, only on the structure of connections. The examples above look like this:
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These are examples of motifs: recurring structures whose value doesn’t depend on who participates, only on the structure of connections. The examples above look like this:
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Usually the values we assign to structures are highly regular. If we encounter the same structure, just with different people, we might give it the same value.
For example, forming a link might have a fixed cost, regardless of who forms the link or who they connect to.
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For example, forming a link might have a fixed cost, regardless of who forms the link or who they connect to.
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October 14, 2025 at 4:18 PM
Usually the values we assign to structures are highly regular. If we encounter the same structure, just with different people, we might give it the same value.
For example, forming a link might have a fixed cost, regardless of who forms the link or who they connect to.
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For example, forming a link might have a fixed cost, regardless of who forms the link or who they connect to.
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There is still a complication in our analysis of the network formation process. Even if we find the potential, the space of networks is huge. For example, there are more networks with 20 than atoms in the universe! If we want interpretable results, we need to do better.
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October 14, 2025 at 4:18 PM
There is still a complication in our analysis of the network formation process. Even if we find the potential, the space of networks is huge. For example, there are more networks with 20 than atoms in the universe! If we want interpretable results, we need to do better.
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When the game is a potential game, the stationary distribution has the following form, called a Gibbs measure. What is Φ? It’s the potential of the game!
This means that the long-run properties of the process have a clear relation to the static properties of the game.
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This means that the long-run properties of the process have a clear relation to the static properties of the game.
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October 14, 2025 at 4:18 PM
When the game is a potential game, the stationary distribution has the following form, called a Gibbs measure. What is Φ? It’s the potential of the game!
This means that the long-run properties of the process have a clear relation to the static properties of the game.
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This means that the long-run properties of the process have a clear relation to the static properties of the game.
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Let’s go back to your social network. Over time you (and everyone else) will face the choice of forming or severing friendships many times. This gives rise to a Markov chain of networks, whose stationary distribution tells us about the long-run behavior of the process.
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October 14, 2025 at 4:18 PM
Let’s go back to your social network. Over time you (and everyone else) will face the choice of forming or severing friendships many times. This gives rise to a Markov chain of networks, whose stationary distribution tells us about the long-run behavior of the process.
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What happens when everyone values a structure equally? Then the game is a potential game. This means that there exists a single function (the potential) that captures the incentives to deviate of all players. Focusing on potential games greatly simplifies our analysis.
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October 14, 2025 at 4:18 PM
What happens when everyone values a structure equally? Then the game is a potential game. This means that there exists a single function (the potential) that captures the incentives to deviate of all players. Focusing on potential games greatly simplifies our analysis.
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I show that these two approaches are equivalent, and how to switch between the two. Thinking about sub-structures turns out to be particularly useful, since they capture the incentives to change the network. This makes analyzing things like equilibria much cleaner.
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October 14, 2025 at 4:18 PM
I show that these two approaches are equivalent, and how to switch between the two. Thinking about sub-structures turns out to be particularly useful, since they capture the incentives to change the network. This makes analyzing things like equilibria much cleaner.
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Adding this friendship will change how much you value the social network. You can think about your choice in two ways:
1. How would this friendship change your valuation of the whole network?
2. Which new sub-structures does this link create, and how much do you value them?
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1. How would this friendship change your valuation of the whole network?
2. Which new sub-structures does this link create, and how much do you value them?
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October 14, 2025 at 4:18 PM
Adding this friendship will change how much you value the social network. You can think about your choice in two ways:
1. How would this friendship change your valuation of the whole network?
2. Which new sub-structures does this link create, and how much do you value them?
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1. How would this friendship change your valuation of the whole network?
2. Which new sub-structures does this link create, and how much do you value them?
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Networks are fluid objects. Regardless of the application you’re interested in, chances are links are being created and destroyed all the time.
Let’s think about your social network. You might bump into someone new and think about whether you want to be their friend.
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Let’s think about your social network. You might bump into someone new and think about whether you want to be their friend.
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October 14, 2025 at 4:18 PM
Networks are fluid objects. Regardless of the application you’re interested in, chances are links are being created and destroyed all the time.
Let’s think about your social network. You might bump into someone new and think about whether you want to be their friend.
2/
Let’s think about your social network. You might bump into someone new and think about whether you want to be their friend.
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This is great! I’d like to join too, will share new work soon
November 19, 2024 at 4:51 AM
This is great! I’d like to join too, will share new work soon