Distributions of centrality on networks

Abstract

We provide a framework for determining the centralities of agents in a broad family of random networks. Current understanding of network centrality is largely restricted to deterministic settings, but practitioners frequently use random network models to accommodate data limitations or prove asymptotic results. Our main theorems show that on large random networks, centrality measures are close to their expected values with high probability. We illustrate the economic consequences of these results by presenting three applications: (1)In network formation models based on community structure (called stochastic block models), we show network segregation and differences in community size produce inequality.Benefits from peer effects tend to accrue disproportionately to bigger and better-connected communities.(2) When link probabilities depend on spatial structure, we can compute and compare the centralities of agents in different locations. (3) In models where connections depend on several independent characteristics, we give a formula that determines centralities 'characteristic-by-characteristic'. The basic techniques from these applications, which use the main theorems to reduce questions about random networks to deterministic calculations, extend to many network games.In many settings of economic interest, agents benefit from connections to others. These peer effects depend on network structures, and better positioned agents can benefit much more than their less central counterparts. In education, for example, students form networks of friends and these connections affect academic achievement through group study, as a source of motivation, etc. Empirical evidence suggests that the impact of these peer effects on outcomes is approximated well by measures of network centrality such as Katz-Bonacich centrality (Calvó-Armengol, Patacchini, and Zenou (2009), Hahn, Islam, Patacchini, and Zenou (2015)). More generally, measures of centrality and related quantities are crucial to understanding economic models from peer effects and quadratic games on networks to social learning models such as DeGroot updating. 1 While there is a large literature on Katz-Bonacich and other centrality measures in deterministic settings, relatively little is known about centrality measures on stochastic networks.But in many applied settings, precise data about the full network is not available. Researchers instead use statistical models of network formation where links form with probabilities depending on agent characteristics. As a very simple example, one could model the social network in a school with black and white students by assuming two students of the same race are friends with probability 50% while two students of different races are friends with probability 25% (where all connections form independently). Moreover in theoretical work, varying parameters in models of random network formation often provides more insight than comparing particular deterministic networks.The current paper gives a framework for determining how central eac...

Targeted Intervention in Random Graphs

“…As noted in [16], implicit from the proof of Theorem 6 in [15] (where it is Theorem 1) is a deviation bound on the matrix norm. The result holds for all random graphs with independent edges, and we restate it here for the special case of G(w) graphs.…”

“…We make use of concentration inequalities for the spectra of random adjacency matrices; there is a great deal of work studying various spectral properties of random graphs (see e.g. [14,15,13,2,16]). Particularly relevant to us is [16], which characterizes the asymptotic distributions of various centrality measures for random graphs.…”

“…[14,15,13,2,16]). Particularly relevant to us is [16], which characterizes the asymptotic distributions of various centrality measures for random graphs. There is further relevant literature for studying centrality in graphons, see e.g.…”

Resource sharing on endogenous networks

“…In the case of endogenous formation this assumption may not be satisfied, rendering spectral methods difficult to implement outside of particular special cases involving links that are fully independent (e.g. Dasaratha, 2020).…”

Graphon Games: A Statistical Framework for Network Games and Interventions

“…The concurrent work [40] suggests the use of graphons to extend the setup of mean-field games (which differently from network games are dynamic and stochastic games) to heterogeneous settings. Finally, we remark that the idea of interpreting observed graphs as random realizations from an underlying random graph model has recently been used in the study of centrality measures in [41] for stochastic block models and in [33] for graphon models. The authors of these papers study among others Bonacich centrality, which is known to coincide with the equilibrium of a specific type of network games (with scalar nonnegative strategies, quadratic payoff functions and strategic complements).…”