Yann Bramoullé, Garance Genicot, 2024, Journal of Economic Theory, Volume 222, pp. 105920.
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We show, in particular, that diffusion centrality relies on diffusion processes where individuals independently transmit all the messages received in the previous period to each of their neighbors.
RESEARCH QUESTION
Diffusion in networks plays a critical role in our interconnected societies, underlying the propagation not only of germs and technologies, but also of ideas, misinformation, and data. In an important study, Banerjee et al. (2013) introduced a new metric termed diffusion centrality. Its aim is to quantify the extent to which a given piece of information is disseminated among agents within a network over time. This notion has become a cornerstone of research on diffusion. For instance, diffusion centrality performs well, empirically, in explaining take-up rates of a microfinance loan program in rural India. Diffusion centrality has also been used to analyze voting behavior, in contexts where citizens appeal for favors from politicians.
Garance Genicot, a professor at Georgetown, has been a regular visitor at AMSE. During one of her first visits, we started to work together on diffusion in networks. We quickly realized two important facts about diffusion centrality, and the way researchers were using this metric. First, its precise theoretical foundations were somewhat ambiguous, with only partial descriptions of the models microfounding the notion. This was a source of confusion. Second, diffusion centrality relied on assumptions which were sometimes at odds with the specific contexts in which it was applied. In the voting application, in particular, the shared information entails statements like “agent i requires a favor from politician j”. Diffusion centrality assumes that this information is retransmitted by the politician and to the agent during diffusion, an unrealistic assumption. We investigated these two issues in some depth, which eventually led to this publication.
PAPER’S CONTRIBUTIONS
Our paper offers two main contributions. First, we characterize the precise theoretical foundations of diffusion centrality. We show, in particular, that this metric relies on diffusion processes where individuals independently transmit all the messages received in the previous period to each of their neighbors. Messages received before the previous period are ignored, and messages received multiple times in the previous period are transmitted the same number of times.
Second, we consider alternative diffusion processes where the sender or the receiver, or both, do not retransmit information during diffusion. These processes may be more appropriate to model situations of bullying, as victims of bullying will likely not spread hurtful rumors about themselves, targeted requests, and political intermediation.
We call targeting centrality the centrality metric emerging from these alternative models. We were able to derive explicit analytical formulas for targeting centrality, with both finite and infinite time horizons. These formulas rely on elementary operations over powers of the network’s adjacency matrix, as with diffusion centrality. We also analyzed the differential effects of removing retransmission by the sender only or by the receiver only, showing that these two assumptions have opposite effects on the relationship between targeting and diffusion centrality in a limit scenario with an infinite time horizon and when the probability of information transmission is high.
POLICY IMPLICATIONS
Introducing no-retransmission by one or two agents may appear to be a small change in the modeling framework. We show, however, that this change actually has large effects on centrality comparisons. The Figure depicts, for instance, the correlation coefficient between targeting and diffusion centrality across agents within a network, when both the sender and the receiver do not retransmit, for many realizations of random graphs. As the probability of information transmission increases, this correlation decreases and can even become negative on some structures. Results are similar when looking at correlation in ranks. Targeting and diffusion centrality thus generally provide fairly different ranking of nodes. This may have important policy implications, both in contexts where we want to maximize diffusion, as with the adoption of better technologies and practices, and where we want to minimize it, as with targeted vaccinations.
REFERENCE
Banerjee Abhijit, Chandrasekhar Arun G., Duflo Esther, Jackson Matthew O., 2013, The diffusion of microfinance, Science 341.
→ This article was issued in AMSE Newletter, Winter 2024.
