How to determine 'burstiness' of data? This is both a Math- and an R-question. I have a vector of POSIXct dates and a I want to determine the characteristics of the data.
Example with numbers:
c1 = 1,2,50,51,100,101,120,121
c2 = 10,20,30,40,50,60

c1 has more jumps and c2 is more continuous, but I am not sure how to sum this up in one value. Any ideas?
Consider a waiting queue, and the arrival dates as elements of the vector. Now I want to say: f(c1) < f(c2) for a function f to determine the smoothness of the arrival dates.
It is all about video uploads, in which I want to determine a value, whether there is a continuous frequency of uploads (daily upload) or a more bursty behaviour (5 videos on Monday and the next video on Friday)
 A: (This is an expansion on whuber's comment.)
Your data are arrival times, which are commonly summarized by the waiting time distribution (i.e. time difference between consecutive arrivals).
For example, a simple single-number summary might be the coefficient of variation of the arrival times. For the standard "memoryless" null model, the arrival time PDF would be exponential, with a unit coefficient of variation.
(This would be comparable to the "index of dispersion" approach discussed here, which applies to count data, # arrivals in a given time.)
A: +1 to @geomatt22 and whuber for their comments and suggestions. Their suggestions work when the information (jumps, first differences, etc.) is not extreme valued. A recent paper by Lin and Tegmark Critical Behavior from Deep Dynamics: A Hidden Dimension in Natural Language (available here ... https://ai2-s2-pdfs.s3.amazonaws.com/5ba0/3a03d844f10d7b4861d3b116818afe2b75f2.pdf), discusses situations that frequently occur where the information is power-lawed, extreme valued and exhibits critical complexity. In particular, they take "traditional" sequential analysis based on Markov processes to task for being "shallow" and unable to capture deep, long-term correlations. Here is their abstract:

We show that in many data sequences — from texts in different
  languages to melodies and genomes — the mutual information between two
  symbols decays roughly like a power law with the number of symbols in
  between the two. In contrast, we prove that Markov/hidden Markov
  processes generically exhibit exponential decay in their mutual
  information, which explains why natural languages are poorly
  approximated by Markov processes. We present a broad class of models
  that naturally reproduce this critical behavior. They all involve deep
  dynamics of a recursive nature, as can be approximately implemented by
  tree-like or recurrent deep neural networks. This model class captures
  the essence of probabilistic context-free grammars as well as
  recursive self-reproduction in physical phenomena such as turbulence
  and cosmological inflation. We derive an analytic formula for the
  asymptotic power law and elucidate our results in a statistical
  physics context: 1-dimensional “shallow” models (such as Markov models
  or regular grammars) will fail to model natural language, because they
  cannot exhibit criticality, whereas “deep” models with one or more
  “hidden” dimensions representing levels of abstraction or scale can
  potentially succeed.

This isn't intended to suggest that GeoMatt22 and Whuber's suggestions are wrong, it's merely intended to suggest an alternative formulation of the problem. 
In addition to Tegmark's work, another recent paper by J.P. Bouchaud Crises and Collective Socio-Economic Phenomena (available here ... https://www.cfm.fr/assets/ResearchPapers/Crises+and+collective+socio-economic+phenomena.pdf) specifically discusses modeling behaviors such as sudden ruptures, crises and avalanches, which are close analogues to burstiness. Here is their abstract:

Financial and economic history is strewn with bubbles and crashes,
  booms and busts, crises and upheavals of all sorts. Understanding the
  origin of these events is arguably one of the most important problems
  in economic theory. In this paper, we review recent efforts to include
  heterogeneities and interactions in models of decision. We argue that
  the so-called Random Field Ising model (rfim) provides a unifying
  framework to account for many collective socio-economic phenomena that
  lead to sudden ruptures and crises. We discuss different models that
  can capture potentially destabilising self-referential feedback loops,
  induced either by herding, i.e. reference to peers, or trending, i.e.
  reference to the past, and that account for some of the phenomenology
  missing in the standard models. We discuss some empirically testable
  predictions of these models, for example robust signatures of
  rfim-like herding effects, or the logarithmic decay of spatial
  correlations of voting patterns. One of the most striking result,
  inspired by statistical physics methods, is that Adam Smith’s
  invisible hand can fail badly at solving simple coordination problems.
  We also insist on the issue of time-scales, that can be extremely long
  in some cases, and prevent socially optimal equilibria from being
  reached. As a theoretical challenge, the study of so-called
  “detailed-balance” violating decision rules is needed to decide
  whether conclusions based on current models (that all assume
  detailed-balance) are indeed robust and generic.

Together, these two papers represent significant advances in the analysis of extreme valued behaviors.
To be specific wrt your question about developing a "single" value to compare the two distributions, there are certainly many ways to do this. One way that incorporates the possibility of the information being extreme valued is to estimate the tail index of the distribution -- whether raw data, first differences, whatever. One easily generated approach to tail estimation is explained in Gabaix's paper on OLS modeling of the log-ranks (available here ... http://www.eco.uc3m.es/temp/jbes.2009.06157.pdf) or by leveraging the more rigorous and computationally intensive methods developed by Pickands or Hill. Once an index is available, then a distribution can be assigned based on the Examples section of this Wiki discussion of the Tweedie family of distributions... https://en.wikipedia.org/wiki/Tweedie_distribution
