We model scientific theories as Bayesian networks. Nodes carry credences and function as abstract representations of propositions within the structure. Directed links carry conditional probabilities and represent connections between those propositions. Updating is Bayesian across the network as a whole. The impact of evidence at one point within a scientific theory can have a very different impact on the network than does evidence of the same strength at a different point. A Bayesian model allows us to envisage and analyze the differential impact of evidence and credence change at different points within a single network and across different theoretical structures.

Our scientific theories, like our cognitive structures in general, consist of propositions linked by evidential, explanatory, probabilistic, and logical connections. Those theoretical webs ‘impinge on the world at their edges,’ subject to a continuing barrage of incoming evidence (Quine 1951, 1953). Our credences in the various elements of those structures change in response to that continuing barrage of evidence, as do the perceived connections between them. Here we model scientific theories as Bayesian nets, with credences at nodes and conditional links between them modelled as conditional probabilities. We update those networks, in terms of both credences at nodes and conditional probabilities at links, through a temporal barrage of random incoming evidence. Robust patterns of punctuated equilibrium, suggestive of ‘normal science’ alternating with ‘paradigm shifts,’ emerge prominently in that change dynamics. The suggestion is that at least some of the phenomena at the core of the Kuhnian tradition are predictable in the typical dynamics of scientific theory change captured as Bayesian nets under even a random evidence barrage.

We discuss spectral clustering from a variety of perspectives that include extending techniques to rectangular arrays, considering the problem of discrepancy minimization, and applying the methods to directed graphs. Near-optimal clusters can be obtained by singular value decomposition together with the weighted k k -means algorithm. In the case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, while in the case of edge-weighted graphs, a normalized Laplacian-based clustering. In the latter case, it is proved that a spectral gap between the ( k − 1 ) \left(k-1) st and k k th smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is 2 k − 1 {2}^{k-1} , but only the first k − 1 k-1 eigenvectors are used in the representation. The ensemble of these eigenvectors constitute the so-called Fiedler-carpet.

Spectral clustering is discussed from many perspectives, by extending it to rectangular arrays and discrepancy minimization too. Near optimal clusters are obtained with singular value decomposition and with the weighted k-means algorithm. In case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, and in case of edge-weighted graphs, a normalized Laplacian based clustering. In the latter case it is proved that a spectral gap between the (k − 1)th and kth smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is 2 k−1 , but only the first k − 1 eigenvectors, constituting the so-called Fiedler-carpet, are used in the representation. Application to directed migration graphs is also discussed.