Threshold Phenomena in Epistemic Networks

A small consortium of philosophers has begun work on the implications of epistemic networks (Zollman 2008 and forthcoming; Grim 2006, 2007; Weisberg and Muldoon forthcoming), building on theoretical work in economics, computer science, and engineering (Bala and Goyal 1998, Kleinberg 2001; Amaral et. al., 2004) and on some experimental work in social psychology (Mason, Jones, and Goldstone, 2008). This paper outlines core philosophical results and extends those results to the specific question of thresholds. Epistemic maximization of certain types does show clear threshold effects. Intriguingly, however, those effects appear to be importantly independent from more familiar threshold effects in networks. 1. Epistemology and Scientific Networks Epistemology is defined as the study of knowledge. The traditional focus in the field, however, has long been limited to the study of the individual epistemic agent. Traditional epistemology treats knowledge acquisition as an individual endeavor. In Hume, Descartes, and Kant, epistemology is told as the story of a single individual trying to figure out what the world is like—an attempt to answer the question of how an individual agent figures out the structure of the world. A small consortium of contemporary philosophers has begun work on a different approach (Zollman 2008 and forthcoming; Grim 2006, 2007; Weisberg and Muldoon forthcoming). In this recent work the essential emphasis is not on communities of epistemic agents rather than on the individual. How does an individual figure out the structure of the world? The truth is that no individual does. It is cultures and communities that plumb the structure of reality; individuals figure out the structure of the world only as they participate in the epistemic networks in which they are embedded. Science is undoubtedly our pre-eminent example of knowledge acquisition. But what characterizes contemporary scientific research is not a catalog of isolated investigators but coordinated interactive networks of investigation. To understand knowledge acquisition in science one must understand more than the work of individual participants. One must understand the structure and dynamics of the enterprise as whole. Here questions are importantly different than those in traditional epistemology. A scientific community can be envisaged as a network of interactive agents attempting to limn reality on the basis of uneven, conflicting, and sometimes ambiguous data. How does the network structure of collaboration and competition, of data sharing and information transfer, affect knowledge acquisition in the community at large? What kinds of network structures, of what kinds of agents, will best achieve scientific goals— scientific goals of accuracy, for example? In what ways will those structures be sensitive to the specific form of the problem, or to the distribution or uncertainty of data? Those are questions central to this new approach, and questions for which the work outlined below offers some early and partial clues. Given what we know of networks in general, it is to be expected that the dynamics of information acquisition and exchange across an epistemic network will not be reducible to any simple aggregate measure across individuals. The modeling results offered here substantiate that expectation in full. One of the implications of epistemic networks, tracked here in terms of thresholds, is the robust and surprising finding that a scientific community may learn more when its individual scientists learn less. In terms of central scientific goals such as accuracy, increased informational linkages between scientists may not always be a good thing. 17 th century science was characterized by distributed informational networks with limited linkages between investigators. 21 st century science is characterized by totally connected networks across the internet. One way of phrasing a central result in what follows is that for some central scientific goals, including accuracy, and for some topics of investigation, the network structure of 17 th century science appears to be superior to our own. Section 2 outlines the notion of epistemic landscape crucial to the model, with details in section 3 of initial networks surveyed. The core result that a scientific community can learn more when individual scientists learn less is presented in section 4. Sections 5 and 6 further explore the question of precisely what properties of networks are important for that result. Here results show clear thresholds for epistemic maximization of certain types with increasing number of links in random networks. Epistemic maximization on networks of the type at issue, it turns out, exhibits clear threshold phenomena. But it also turns out that the epistemic thresholds at issue are surprisingly independent from other network; they do not correlate cleanly with thresholds in any of the other network properties one might expect. Results here are intended as an introduction, with first hints regarding some of the surprises and subtleties of informational dynamics across epistemic networks. These are offered as a first word on the topic, rather than the last word; it quickly becomes clear how much we do not yet understand, and how much more work remains to be done. 2. Epistemic Landscapes We can envisage an epistemic landscape as a topology of ideal data—data regarding alternative medical treatments, for example (Fig. 1). In such a graph points in the xz plane represent particular combinations of radiation and chemotherapy, or particular hypotheses regarding the best combination. Graph height on the y axis represents some measure of success: the proportion of patients in fact cured with combinations of radiation and chemotherapy at that rate. If you use radiation therapy at rate x, and chemotherapy at rate z, you will get the proportion of cures represented on the y axis hovering over that point. Fig. 1 A three-dimensional epistemic landscape. Points on the xz plane represent hypotheses regarding optimal combination of radiation and chemotherapy; graph height on the y axis represents some measure of success. This first epistemic landscape is a medical one, but the specific topic of investigation is unimportant for our broader epistemic concerns. One might have an epistemic landscape of magnetology readings for different hypothetical locations of a shipwreck, or of irridium stratigraphy world-wide as feedback regarding different hypotheses regarding the timing of the K-T asteroid collision, or any measurable variable y that confirms some hypotheses rather than others regarding the interplay of variables x and z. It is important to emphasize, however, that the concept of an epistemic landscape represents ideal data across a full range of possible hypotheses. Different investigators will test different hypotheses and will get differential feedback regarding those hypotheses. As an individual investigator, however, one will not be able to see the epistemic landscape as a whole. One will see results only at a point in the graph, in a small area or in a scattering of points. Despite those limitations, our job description as epistemic agents is to find the theory that is best supported by data. The goal of investigation is to find the highest points in the epistemic landscape—the best confirmed hypotheses, or the most warranted predictions, the most reliable medical treatments. Fortunately, we do not work alone: we are linked to other investigators as part of a larger network. The model at issue here employs simpler twodimensional epistemic landscapes (Fig. 2). Fig. 2 Two-dimensional epistemic landscapes. Values on the x axis represent alternative hypotheses. Values on the y axis represent the ideal epistemic payoff for particular hypotheses. In the first landscape data converges smoothly to a single best hypothesis or medical treatment. The second represents a slightly more complex landscape, in which particular combinations of drugs do well, perhaps, but combinations in between do worse. The third is a still more complex landscape, in which some peaks are smooth and easily climbed, but represent inferior outcomes. The hypotheses or medical treatments they lead to are not the best. The best outcome, however—that hypothesis that would be best confirmed, or that medical treatment that would be most effective—is hidden in a spike with a narrow base, and is thus harder to find. 3. Modeling Epistemic Networks Suppose we have a population of agents, each of whom starts with a hypothesis. Here that hypothesis is represented by a single point on the x-axis landscape. In testing their hypotheses, our agents accumulate data as feedback—a percentage of patient cures, for example. But our agents are also networked to others; they can see not only the success rate of their own hypothesis but the success rate for the hypotheses of those to whom they are linked. Agents change their hypotheses based on the success rates of those to whom they are linked. As an agent in this model, you can see how well the hypotheses of some other agents are doing; if their hypotheses are better supported by the data than yours, you shift your hypothesis in their direction. If your hypothesis is the best of those visible to you, on the other hand, you stick with it. With even a network model this simple there are a number of intriguing parameters. One of the built into this model is a ‘shaking hand’: when you aim to duplicate another’s hypothesis, you may be slightly off. Your lab conditions may be slightly different the other agent, or your chemicals impure, or your sample slightly biased. You therefore end up with a hypothesis that is not a precise match of that you are imitating but is merely close by. One result, of course, is that you therefore explore more of the epistemic landscape. The model used here builds in a ‘shaking hand’ that puts one in random region within 4 points either side of a target hypothesis. The model also incorporates elements of ‘speed’ and ‘inertia’. In pursuing a more successful hypothesis, does one jump to that conclusion or approximate it halfway each time? This model employs the latter assumption, with a ‘speed’ of 50%. It also builds in an ‘inertia’ factor of 50%, representing agents’ stubborn investment in hypothesis. In each of 100 steps each agent has only a 50% probability of shifting in the direction of a superior hypothesis. The crucial parameter the model is designed to investigate, of course, is network structure (Fig. 3). of an epistemic