Change is the very nature of interaction patterns in biology, technology, economy, and science itself: The interactions within and between organisms change; the air, ground, and sea traffic change; the global financial flow changes; and the scientific research front changes. With increasingly available data, networks and clustering tools have become important tools to comprehend instances of these large-scale structures. But blind to the difference between noise and trends in the data, these tools alone must fail when used to study change. Only if we can assign significance to the partition of single networks can we distinguish structural changes from fluctuations and assess how much confidence should we have in the changes. Here we show that bootstrap resampling accompanied by significance clustering provides a solution to this problem. We use the significance clustering to realize de Solla Price's vision of mapping the change in science. Network analysis provides tools for understanding social and biological systems with numerous and diverse interacting components. For large networks, we need ways to highlight the important features while simplifying the overall structure. Researchers have developed a suite of network mapping tools for this purpose (1, 2, 3, 4); with them we can abstract, quantify, and comprehend the nature of a complex systems. Powerful as these tools have proven for understanding a system's structure, we do not yet have an adequate tool for mapping how this structure changes. For example: How does the organization of social contacts change when diseases develop and spread? How does the network structure of the federal funds market change when credit markets freeze up? How do gene regulatory networks differ between cancer and non-cancer states? How has the network of global air traffic changed over the past half century? And how does science itself evolve as paradigms shift through time? Any tool for analyzing change must distinguish between meaningful trends and statistical noise. For example, statistical network models and stratified data make it possible to estimate global properties, and the associated level of confidence, of large networks from observation of sample networks (5, 6, 7). But this comes at the cost of losing the unique identity of individuals. The reason that recent network approaches have become so prominent in the study of complex systems is that they capture and respect the identities and characteristics of the components (8, 9). Often these individual differences matter critically -and clustering rather than stratification must be depolyed to comprehend the data (1, 2, 3, 4, 10 To assign significance to clusters of single networks, the bootstrap method is compelling (15). The bootstrap is a method for assessing the accuracy of an estimate by resampling from the empirical distribution of observations. But what do we do if we have only one observation, a single network? When the single observation is composed of numerous components, as is a network of nodes and links, the parametric bootstrap provides a solution. Instead of resampling directly from the empirical distribution, a parametric model is used to fit the data. For the networks of interest here, the identities of the nodes cannot be altered, parametrized, or resampled -it makes no sense to talk about the US air transit network without O'Hare, let alone with two O'Haresbut the link weights, which effectively define the nodes, can be parametrized and resampled without undermining the individual characteristics of the nodes. With this approach we can assess the significance of clusters and estimate the accuracy of summary statistics, based on the proportion of bootstrap networks that support the observation (see Fig. 1 The standard approach to cluster networks is to minimize an objective function over possible partitions of the network as in the left side of the diagram. By repeatedly resampling of the weighted links from the original network, we create a "bootstrap world" of resampled networks. By clustering these as well, and comparing to the clustering of the original network, we can estimate the degree of support that the data provide in assigning each node to a cluster. In the bottom network, the darker nodes are clustered together in at least 95% of the 1000 bootstrap networks. change in the structure of science itself (16). Science is a dynamic, organized, and massively parallel human endeavor to discover, explain, and predict the nature of the physical world. In science, new ideas are built upon old ideas, and through cumulative cycles of modeling and experimenting scientific research undergoes constant change: fields grow and shrink, merge and split. Citation patterns among scientific journals allow us to glimpse this flow of ideas and how the flow of ideas changes over time (16). Here we use journal aggregated citation data (18) the networks with the information theoretic clustering method presented in ref. (4), which can reveal regularities of information flow across directed and weighted networks. We emphasize that, with the appropriate modification, the method of bootstrap resampling accompanied by significance clustering presented here is general and works for any type of network. To assess the accuracy of a clustering, we resample a large number B ∼ 1000 of bootstrap networks from the original network. For the directed and weighted citation 3 network of science, we treat the citations as independent events and resample the weight of each link from a Poisson distribution with mean as in the original network (19) (see for example refs. (20, 21, 22) for other resampling techniques). Given the original network, Fig. 1 illustrates the clustering of this network and the clusterings of four of the bootstrap networks. When dealing with scalar summary statistics it is straightforward to assign a 95% bootstrap confidence interval as spanning the 2.5th and 97.5th percentiles of the boostrap distribution, but to assess the accuracy of the clusters requires a different approach. To identify the nodes that are significant in their cluster assignments, we use simulated annealing to search for the largest significant subset of nodes within each cluster of the original network that are clustered together in at least 95% of all bootstrap networks. To identify the clusters that are significantly distinct from any other cluster, we search for all clusters whose significant subset are clustered with no other cluster's significant subset in at least 95% of all bootstrap networks (see Method section). The first step of Fig. 2 illustrates this process as applied to a network at two different time points. Once we have a significance cluster for the network at each time point (or each state) we need to simplify and highlight the structural changes between clusters and thereby bring out the stories in our data. In the second step of Fig. 2 , we show how to construct an alluvial diagram that highlights and summarizes the structural changes between the time 1 and time 2 significance clusters. Each colored cluster in the network is represented by an equivalently colored block in the alluvial diagram. Solid colors represent significantly assigned nodes, while lighter colors represent insignificant assignments. Changes in the clustering structure from one time period to another are represented by the mergers and divergences that occur in the ribbons linking the blocks at time 1 and time 2. To illustrate the power of this approach, we apply this method to citation data from Thomson-Reuters' Journal Citation Reports. These data aggregate, at the journal level, approximately 60,000,000 citations among more than 7000 journals over the past decade. Comparing the significance clusters for each year by means of an alluvial diagram, we reveal the significant structural changes that have occurred in science over the past decade. Rather than viewing the entire diagram, let us pull out a couple of interesting stories. Fig. 3 show a subset of medical fields for the years 2001, 2003, and 2005. As an illustrative example we describe the gradual integration of nephrology into medicine (purple stream in the diagram). In 2001, the field lead by Kidney International, Transplantation, and Journal of the American Society of Nephrology consists of 39 journals of which 33 are in the significant subset (89% of the citation flow) and the field as a whole is separated from all other fields in 99% of the clustered bootstrap networks. In the alluvial diagram, this is illustrated as a 89% dark purple block in The integration of nephrology is just one of many changes over this period. In the same diagram, we also highlight the biggest change in science over the past decade: the transformation of neuroscience from interdisciplinary specialty to a mature and stand-alone discipline, comparable to physics or chemistry, economics or law, molecular biology or medicine. In 2001 the majority of neuroscience journals (dark orange) are assigned with statistical significance to the field of molecular and cell biology. Others appear in psychology (green) and neurology (blue). In 2003, many of these journals (light orange) remain in molecular and cell biology, but their assignment to this field is no longer significant. The transformation is underway. In 2005, neuroscience first emerges an independent discipline (red). The journals from molecular biology split off completely from their former field and have merged with neurology and a subset of psychology into the stand-alone field of neuroscience. (In 2006, not shown, the structure reverts to a pattern similar to 2003. It will be telling to observe what happens in 2007.) Neuroscience, which originated in the first studies of the nervous system more than a century ago and which for a long time existed as a set of independent disciplines, has now become a unified field. In their citation behavior, neuroscientists have cleaved from their traditional disciplines and united to form what is now the fifth largest field in the sciences of science (after molecular and cell biology, physics, chemistry, and medicine). Although this interdisciplinary integration has been ongoing since the 1950s (23), only in the last decade has this change come to dominate the citation structure of the field and overwhelm the intellectual ties along traditional departmental lines. The problem of detecting structural change in large networks adds two new challenges in addition to the basic problem of network clustering: (1) we need appropriate statistical methods to identify significant features of network clustering and to distinguish between trends and noise in the data, and (2) we require effective visualizations to bring out the stories implicit in a time series of cluster maps. To resolve the first of these challenges, we have developed a method for significance clustering based on the parametric bootstrap. To address the second, we have presented the visualization technique of al- luvial diagrams. These method is general to many types of networks and can be applied to answer questions about structural change in science, economics, and business.
Alluvial diagrams REF reveal how network structures change over time.
Mapping change in large networks
PLoS ONE 5(1): e8694 (2010)
Biology Physics Medicine