Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Abstract

We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

Nonlocal cross-interaction systems on graphs: Nonquadratic Finslerian structure and nonlinear mobilities

“…[21]. However, as was pointed out in [34], in the absence of diffusion, this choice does not allow for an increase of the support of the solution thus a different choice must be made to obtain physically reasonable solutions. Similar problems occur with the geometric mean 𝜃 g (𝑟, 𝑠) = √ 𝑟 𝑠, while dynamics using the arithmetic mean 𝜃 a (𝑟, 𝑠) = 𝑟 +𝑠 2 as an interpolation function may lead to negative densities and is therefore also not a reasonable choice.…”

“…Similar problems occur with the geometric mean 𝜃 g (𝑟, 𝑠) = √ 𝑟 𝑠, while dynamics using the arithmetic mean 𝜃 a (𝑟, 𝑠) = 𝑟 +𝑠 2 as an interpolation function may lead to negative densities and is therefore also not a reasonable choice. However, it is known that for pure transport equations, upwind schemes yield to stable and structure preserving discretizations which eventually motivated Esposito et al [34] to choose 𝜃 𝑗 (𝑟, 𝑠) = 𝑟1 { 𝑗>0} (𝑟, 𝑠) + 𝑠1 { 𝑗<0} (𝑟, 𝑠)…”

“…This comes at the price of losing the negative homogeneity of the action density, obtaining a Finslerian structure instead of a Riemannian structure. As shown in [34], this structure is sufficient to define the notion of gradient flows as curves of maximal slope and leads to stability of gradient flows under narrow convergence, obtaining existence and uniqueness of gradient flows for a large set of base measures 𝜇 via approximation with finite graphs.…”

“…al. [34] that establishes a graph analogy of the aggregation equation, i.e, for a single species. In their work, it is shown that an appropriate definition of the geometry of the underlying space allows to understand a class of interaction equations on graphs as gradient flows, albeit in a Finslerian framework rather than the usual Riemannian setting.…”

“…We provide a rigorous interpretation of the interaction system as a gradient flow in a Finsler setting, extending the recent work of Esposito et. al.,[34], to the case of multiple species. Additionally, we provide a characterization of critical points of the corresponding energy functional which are also stationary state of the dynamics.…”