We model the arrival of mid-price changes in the E-Mini S&P futures contract as a self-exciting Hawkes process. Using several estimation methods, we find that the Hawkes kernel is power-law with a decay exponent close to −1.15 at short times, less than ≈ 10 3 seconds, and crosses over to a second power-law regime with a larger decay exponent ≈ −1.45 for longer times scales in the range [10 3 , 10 6 ] seconds. More importantly, we find that the Hawkes kernel integrates to unity independently of the analysed period, from 1998 to 2011. This suggests that markets are and have always been close to criticality, challenging a recent study which indicates that reflexivity (endogeneity) has increased in recent years as a result of increased automation of trading. However, we note that the scale over which market events are correlated has decreased steadily over time with the emergence of higher frequency trading.

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We introduce a model-independent approximation for the branching ratio of Hawkes self-exciting point processes. Our estimator requires knowing only the mean and variance of the event count in a sufficiently large time window, statistics that are readily obtained from empirical data. The method we propose greatly simplifies the estimation of the Hawkes branching ratio, recently proposed as a proxy for market endogeneity and formerly estimated using numerical likelihood maximization. We employ our method to support recent theoretical and experimental results indicating that the best fitting Hawkes model to describe S&P futures price changes is in fact critical (now and in the recent past) in light of the long memory of financial market activity.

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are the network average and global clustering coefficients and the number of registered users. The algorithms at hand (1) assume no prior knowledge about the network and (2) access the network using only the publicly available interface. More precisely, this work provides (a) a unified approach for clustering coefficients estimation and (b) a new network size estimator. The unified approach for the clustering coefficients yields the first external access algorithm for estimating the global clustering coefficient. The new network size estimator offers improved accuracy compared to prior art estimators.Our approach is to view a social network as an undirected graph and use the public interface to retrieve a random walk. To estimate the clustering coefficient, the connectivity of each node in the random walk sequence is tested in turn. We show that the error drops exponentially in the number of random walk steps. For the network size estimation we offer a generalized view of prior art estimators that in turn yields an improved estimator. All algorithms are validated on several publicly available social network datasets. ACM Reference Format:Liran Katzir and Stephen J. Hardiman. 2015. Estimating clustering coefficients and size of social networks via random walk.