To study the effect of boundaries on diffusion of new products, we introduce two novel analytic tools: The indifference principle, which enables us to explicitly compute the aggregate diffusion on various networks, and the dominance principle, which enables us to rank the diffusion on different networks. Using these principles, we prove our main result that on a finite line, one-sided diffusion (i.e., when each consumer can only be influenced by her left neighbor) is strictly slower than two-sided diffusion (i.e., when each consumer can be influenced by her left and right neighbor). This is different from the periodic case of diffusion on a circle, where one-sided and two-sided diffusion are identical. We observe numerically similar results in higher dimensions.1 If the graph is undirected, then qi,j = qj,i. 2 qi,j = 0 if there is no edge from i to j. 3 See, e.g., (2.5b) and (2.6b).

Summary We consider obstacle identification using wave propagation. In such problems, one wants to find the location, shape, and size of an unknown obstacle from given measurements. We propose an algorithm for the identification task based on a time‐reversed absorbing condition (TRAC) technique. Here, we apply the TRAC method to time‐dependent linear acoustics, although our methodology can be applied to other wave‐related problems as well, such as elastodynamics. There are two main contributions of our identification algorithm. The first contribution is the development of a robust and effective method for obstacle identification. While the original paper presented criteria for accepting or rejecting regions that enclose the obstacle, we use these criteria to develop an algorithm that automatically identifies the location of the obstacle. The second contribution is the utilization of an improved absorbing boundary condition (ABC) for the identification. We use the second‐order Engquist‐Majda ABC, and we implement it with a finite element scheme. To our knowledge, this is the first time that the second‐order Engquist‐Majda ABC is employed with the finite element method, as this boundary condition does not naturally fit in finite element schemes in its original form. Numerical experiments for the algorithms are presented.

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