Networks of Recurrent Events, a Theory of Records, and an Application to Finding Causal Signatures in Seismicity

We propose a method to search for signs of causal structure in spatiotemporal data making minimal a priori assumptions about the underlying dynamics. To this end, we generalize the elementary concept of recurrence for a point process in time to recurrent events in space and time. An event is defined to be a recurrence of any previous event if it is closer to it in space than all the intervening events. As such, each sequence of recurrences for a given event is a record breaking process. This definition provides a strictly data driven technique to search for structure. Defining events to be nodes, and linking each event to its recurrences, generates a network of recurrent events. Significant deviations in properties of that network compared to networks arising from random processes allows one to infer attributes of the causal dynamics that generate observable correlations in the patterns. We derive analytically a number of properties for the network of recurrent events composed by a random process. We extend the theory of records to treat not only the variable where records happen, but also time as continuous. In this way, we construct a fully symmetric theory of records leading to a number of new results. Those analytic results are compared to the properties of a network synthesized from earthquakes in Southern California. Significant disparities from the ensemble of acausal networks that can be plausibly attributed to the causal structure of seismicity are: (1) Invariance of network statistics with the time span of the events considered, (2) Appearance of a fundamental length scale for recurrences, independent of the time span of the catalog, which is consistent with observations of the ``rupture length'', (3) Hierarchy in the distances and times of subsequent recurrences.Comment: 19 pages, 13 figure

Evolution in random fitness landscapes: the infinite sites model

We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution $g(w)$. This is the finite population version of Kingman's house of cards model [J.F.C. Kingman, \textit{J. Appl. Probab.} \textbf{15}, 1 (1978)]. In contrast to Kingman's work, the focus here is on unbounded distributions $g(w)$ which lead to an indefinite growth of the population fitness. The model is solved analytically in the limit of infinite population size $N \to \infty$ and simulated numerically for finite $N$. When the genome-wide mutation probability $U$ is small, the long time behavior of the model reduces to a point process of fixation events, which is referred to as a \textit{diluted record process} (DRP). The DRP is similar to the standard record process except that a new record candidate (a number that exceeds all previous entries in the sequence) is accepted only with a certain probability that depends on the values of the current record and the candidate. We develop a systematic analytic approximation scheme for the DRP. At finite $U$ the fitness frequency distribution of the population decomposes into a stationary part due to mutations and a traveling wave component due to selection, which is shown to imply a reduction of the mean fitness by a factor of $1-U$ compared to the $U \to 0$ limit.Comment: Dedicated to Thomas Nattermann on the occasion of his 60th birthday. Submitted to JSTAT. Error in Section 3.2 was correcte