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Non-Backtracking Random Walks and a Weighted Ihara’s Theorem
We study the mixing rate of non-backtracking random walks on graphs by looking at non-backtracking walks as walks on the directed edges of a graph. A result known as Ihara’s Theorem relates the adjacency matrix of a graph to a matrix related to non-backtracking walks on the directed edges. We prove a weighted version of Ihara’s Theorem which relates the transition probability matrix of a non-backtracking walk to the transition matrix for the usual random walk. This allows us to determine the spectrum of the transition probability matrix of a non-backtracking random walk in the case of regular graphs and biregular graphs. As a corollary, we obtain a result of Alon et. al. in [1] that in most cases, a nonbacktracking random walk on a regular graph has a faster mixing rate than the usual random walk. In addition, we obtain an analogous result for biregular graphs.Mathematic
Lead isotopic evidence for synextensional lithospheric ductile flow in the Colorado River extensional corridor, western United States
This is the published version. Copyright 1998 American Geophysical Union. All Rights Reserved.Temporal changes in the Pb isotopic compositions of Miocene lavas erupted in the northern Colorado River extensional corridor suggest that lithospheric mantle and middle to deep crust migrated from beneath the Colorado Plateau into the corridor during extension. Basaltic to rhyolitic lavas erupted in the extensional corridor prior to 12.2 Ma have Pb isotopic values that are similar to those of Tertiary to Quaternary lavas erupted through Proterozoic Mojave crust, which comprises surface exposures of basement in the corridor and much of the extended territory to the west. In contrast, most post-12.2 Ma lavas from the same region have Pb isotopic compositions similar to those of lavas erupted through Arizona crust, which forms the basement of the Colorado Plateau. The changes in isotopic compositions of the basaltic lavas, and perhaps a portion of the changes in isotopic compositions of silicic lavas, are attributed to a change in the composition of the mantle source. However, the 206Pb/204Pb ratios for lavas erupted before and after 12.2 Ma in the corridor decrease with decreasing MgO concentrations, suggesting that the Pb isotopic compositions of crustal assimilants changed at about the same time as the composition of the mantle. In the area of the Black Mountains accommodation zone, the surface boundary between the Arizona and Mojave crustal provinces lies a minimum of 60–80 km to the east of the westernmost lava with an Arizona Pb isotopic signature. This distance cannot be accounted for by displacements along nearby major faults, suggesting that middle to deep Arizona crust flowed a significant distance to the west during extension