We begin by acknowledging the sheer size of the citation index to date, and then discuss the disparity in citations that these papers receive. These differences in impact among papers can be captured by a citation distribution, which can be approximated by a power-law function. We compare power-law distributions to Gaussian distributions, illustrating the distinctions between the two and what they tell us about citation patterns. We then explore the differences in average number of citations between fields, which can make cross-disciplinary comparisions complicated. Luckily, we find that citation patterns are surprisingly universal relative to the field a paper is published in, which allows us identify common trends in citation and impact regardless of discipline. We end with a discussion of what citations don’t capture, given that they are frequently used as a proxy for impact. We pinpoint some potential flaws in this metric, but see citation patterns as a valuable way to gauge the collective wisdom of the scientific community.

Given the jump–decay citation patterns discussed in the previous chapter, are we forced to conclude that the papers we publish will be relevant for only a few years? We find that while aggregate citations follow a clear pattern, the trajectories of individual citations are remarkably variable. Yet, by analyzing individual citation histories, we are able to isolate three parameters – immediacy, longevity, and fitness – that dictate a paper’s future impact. In fact, all citation histories are governed by a single formula, a fact which speaks the universality of the dynamics that at first seemed quite variant. We end by discussing how a paper’s ultimate impact can be predicted using one factor alone: its relative fitness. We show how papers with the same fitness will acquire the same number of citations in the long run, regardless of which journals they are published in.

We end the book by inviting our readership to broaden the science of science for the benefit of all. By thinking beyond disciplinary barriers and considering the benefits of the science of science in its entirety, we hope that future research will increase the depth and advancement of science.

Here we provide an overview of Part I, introducing the main themes we will address as they relate to the science of careers. We ask what mechanisms drive productivity and impact, how creativity is distributed over the course of a career, and whether a scientist’s highest impact work can tell us anything about the other work they produce.

Here we explore the mechanisms and drivers behind the impact disparity discussed in the previous chapter, focusing on what factors create high-impact papers and what conditions contribute to the lognormal distribution citations follow. We show how a rich-get-richer phenomenon similar to preferential attachement, growth, and fitness all contribute to the impact of a paper. We describe a fitness model that can effectively represent these dynamics, providing insight into how impact is created in science.

We ask if it’s possible to accelerate the advancement of science by applying the science of science to the frontiers of knowledge. Using a robot scientist as an example, we show how it is now possible to close the loop by building machines that can create scientific knowledge. We discuss the implications of this on the future of the discipline. Another way to more efficiently advance science is to generate more fruitful hypotheses. We discuss the Swanson hypothesis, which provides a window into how to hone in on valuable discoveries, allowing for the forecasting of frutiful areas of research. We then explore how the frontiers of science can be traced, allowing scientists to more thoughtfully choose topics that will accelerate collective discovery. Finally, we address some challenges posed by this issue, including the “file drawer problem,” which could be mitigated by a more systemic approach to sharing negative results with colleagues in the discipline. We suggest several ways to incentivize and reward impactful science so that we can efficiently reap its benefits.

We begin by showing that age-specific patterns affect the allocation of funding in science. We then ask if there are age specific patterns that dictate when a scientist does her best work, and show that there are universal trends in the age distribution of great innovation. We offer possible explanations as to why these patterns occur. One explanation, which helps explain why scientists typically reach peak performance in middle age, is the “burden of knowledge” theory. Yet this explanation doesn’t account for the discipline-specific trends in age at peak performance that complicate the picture, which may be accounted for by the type of work produced. Research shows that there are two kinds of innovators–conceptual and experimental–and that each has a different peak. Experimental innovators, who accumulate knowledge through experience, tend to peak later. Conceptual innovators, who apply abstract principles, tend to peak earlier. We end by discussing Planck’s principle, which posits that young and old scientists have differing affinities for accepting new ideas.

Here, we focus on two factors that contribute to a paper’s fitness: novelty and publicity. By measuring the novelty of the ideas shared in a paper, we can explore the link between the originality of the research and its impact. Since new ideas are typically snythesized from existing knowledge, we can assess the novelty of an idea by looking at the number domains from which researchers sourced their ideas and how expected or unexpected the combination of domains are. Evidence shows that rare combinations in scientific publications or inventions are associated with high impact. Yet novel ideas are riskier than conventional ones, frequently resulting in failure. Research indicates that scientists tend to be biased against novelty, making unconventional work more difficult to get off the ground. In order to mitigate risk while maximizing novelty, scientists must balance novelty with conventionality. We then look at the role that publicity plays in amplifying a paper’s impact. We find that publicity, whether good or bad, always boosts a paper’s citation counts, indicating that, even in science, it’s better to receive negative attention than no attention at all.

We begin with an anecdote about the largest team in scientific history, and then discuss the shift toward larger teams more generally. We show that the team size distribution has changed its fundamental shape since the 1950s, shifting from a Poussion distribution to a power law distribution as teams have grown larger. These two mathematical shapes represent different modes in which teams form. An exponential distribution leads to the creation of small “core” teams. A power-law distribution results in “extended” teams, accumulating new members in proportion to the productivity of their existing members. These two modes allow us to create an accurate model of team formation, providing us with insight about how team size affects its survival, longevity, and creation of knowledge. We can then assess some of the benefits and drawbacks of large teams, and explore the different kinds of science large and small teams produce. We show how to quantify the disruption of an idea by creating a disruption index, and explain how levels of disruption reflect team size. We end by discussing the implications of the shift to larger teams in science, making a case for preserving smaller teams.

Using a story of credit allocation gone wrong, we introduce some of the challenges that come with assigning credit to collaborative work in science, espeically given the historicial emphaisis on individual acheivement in our field. We explore traditional methods for indicating ownership of scientific work, particularly the ordering of authors on a paper. While this method for understanding who should get the lion’s share of the credit for a discovery is usually effective, it is complicated by discipline-specific variations in the order of authorship. We also look at how alphabetical ordering of authorship in some fields further complicates the picture, how “guest authors” and “ghost authors” reflect flaws in the credit allocation system, and how bias affects the process adversely. We end with a discussion of the alarming colloboration penalty women economists experience, which illustrates the mishaps that can and do occur as a result of the existing system.

Citations reflect the cumulative nature of science, where new research builds on previous discoveries. While citations have been previously used to condense and signal knowledge, they have been employed more recently to gauge the scientific impact of a particular paper or discovery. Groundbreaking work, the thinking goes, should be highly cited by other scientists. In Part III, we explore how scientific impact is quantified, focusing less on the producers of science and more on the work that is produced.

To provide a framework for understanding the importance of team assembly, we open the chapter with a story about teams of chickens. This leads into a discussion about the “too much talent” effect in a range of arenas. We then discuss the role that diversity plays in scientific teamwork. Studies show that diversity among team members – whether ethnic, international, or institutional – promotes the team’s effectiveness, with ethnic diversity offering the largest boost in impact of the resulting paper. We also define collective intelligence and explore that factors that lead to highly intelligent teams. Finally, by mapping the larger networks team members are involved in, we’ve identified four types of links that influence group effectiveness. By varying the proportion of these types of links within a network, we can see how certain coauthorship patterns will impact a team’s success. Taken together, these results show a strong correlation between a team’s composition and the quality of the work it produces. We end by discussing super-ties, or extremely close working relationships that become scientific partnerships, which yield surprising citation and productivity premiums.

The random impact rule allows us to build a null model of a career. Using the null model, we can examine what a scientific career looks like when its driven by chance alone. We call this the R-model. But the R-model only accounts for differences in productivity, not differences in ability or talent. In response to this discrepancy, we create the Q-model, which assumes that the impact of papers we publish is determined by two factors, luck and a Q parameter unique to each person. We can then calculate how a person’s highest impact paper is expected to change with productivity. We find that the Q-model’s predictions are in excellent agreement with real world data. In fact, the Q parameter alone seems sufficient to explain what differentiates one scientist from another. We also find that it remains relatively stable over the course of a career. We then show how we calculate the Q factor for individual scientists, how we can use their Q factor to predict their impact, and how doing so provides a more accurate forecast of a scientist’s future impact than the h-index can. In the case of great scientists, we see that the Q factor turns luck into a consistently high impact career.