Enabling Asymptotic Truth Learning in a Social Network

TL;DR

This paper study both random orderings and carefully crafted decision orders with respect to the graph topology as well as sufficient or necessary conditions for a graph to support such a good ordering to enable network-wide asymptotic truth learning.

Abstract

Consider a network of agents that all want to guess the correct value of some ground truth state. In a sequential order, each agent makes its decision using a single private signal which has a constant probability of error, as well as observations of actions from its network neighbors earlier in the order. We are interested in enabling \emph{network-wide asymptotic truth learning} -- that in a network of $n$ agents, almost all agents make a correct prediction with probability approaching one as $n$ goes to infinity. In this paper we study both random orderings and carefully crafted decision orders with respect to the graph topology as well as sufficient or necessary conditions for a graph to support such a good ordering. We first show that on a sparse graph of average constant degree with a random ordering asymptotic truth learning does not happen. We then show a rather modest sufficient condition to enable asymptotic truth learning. With the help of this condition we characterize graphs generated from the Erdös Rényi model and preferential attachment model. In an Erdös Rényi graph, unless the graph is super sparse (with $O(n)$ edges) or super dense (nearly a complete graph), there exists a decision ordering that supports asymptotic truth learning. Similarly, any preferential attachment network with a constant number of edges per node can achieve asymptotic truth learning under a carefully designed ordering but not under either a random ordering nor the arrival order. We also evaluated a variant of the decision ordering on different network topologies and demonstrated clear effectiveness in improving truth learning over random orderings.

Figures (6)

• Figure 1: An example of a butterfly network of $k=3$.
• Figure 2: Simulation results for the email-univ network with $q=0.7$ over $300$ iterations.
• Figure 3: Here is an example of a single vertex sequence corresponding to one tuple $(a, h)$ from step 1. The $a-1$ row vertex sequences end at the marked nodes. Then, the vertex sequence in row $a$ starts from the far left and goes to the right one by one until the node $(m-1, a)$. Then, it starts from the far right (node $(m+2^{a-1}, a)$) and walks to the left one by one to the node $(m+1, a)$. Lastly, $(m, a)$ gets to aggregate the signals from $(m-1, a)$ and $(m+1, a)$, which is eventually picked up by vertex sequence in row $a+1$. For a fixed $a$, there are $2^k$ such vertex sequences.
• Figure 4: The simulation results for an ER network with $q=0.7$ and $n=1000$, over $300$ iterations. The median learning rate over 300 iterations increases as the edge probability increases. However, the number of negative information cascades also increases, which is reflected in the polarity of learning as well as the lower mean learning rates.
• Figure 5: The simulation results for a PA model with $q=0.7$ and $k=5$ over $300$ iterations. The two neighbors + high value ordering performs much better than random and arrival orderings.

Theorems & Definitions (42)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Lemma 1: Tail bound of Binomial distribution tsun2020
• Proposition 1
• Proposition 2
• Proposition 3
• Lemma 2
• proof : Proof of \ref{['prop:erdos-learning']}