Partition 3d Model
Cheryll Witting
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A uniform distribution of yttrium-90 (90Y) microspheres throughout the entire liver has always been assumed for dose calculation in treating hepatic tumours. A simple mathematical model was formulated which allows estimation of the activities of a therapeutic dose of 90Y microspheres partitioned between the lungs, the tumour and the normal liver, and hence the radiation doses to them. The doses to the tumour and normal liver were verified by intra-operative direct beta-probing. The percentage of activity shunted to the lung and the tumour-to-normal tissue ratio (T/N) were obtained from gamma scintigraphy using technetium-99m-labelled macroaggregated albumin (MAA) which simulates the 90Y microspheres used in subsequent treatment. The intrahepatic activity was partitioned between the tumour and the normal liver based on the T/N and their masses determined from computerized tomography slices. The corresponding radiation doses were computed using the MIRD formula. The estimated radiation doses were correlated with the doses directly measured using a calibrated beta-probe at laparotomy by linear regression. The radiation doses to the tumour and the normal liver, estimated using the partition model, were close to that measured directly with coefficients of correlation for linear regression: 0.862 for the tumours and 0.804 for the normal liver compartment (P
The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on \(n\) nodes with two equal-sized clusters, with an between-class edge probability of \(q\) and a within-class edge probability of \(p\). Although most of the literature on this model has focused on the case of increasing degrees (ie. \(pn, qn \rightarrow \infty \) as \(n \rightarrow \infty \)), the sparse case \(p, q = O(1/n)\) is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborov based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if \(p = a/n\) and \(q = b/n\), then Decelle et al. conjectured that it is possible to cluster in a way correlated with the true partition if \((a - b)^2 > 2(a + b)\), and impossible if \((a - b)^2 < 2(a + b)\). By comparison, the best-known rigorous result is that of Coja-Oghlan, who showed that clustering is possible if \((a - b)^2 > C (a + b)\) for some sufficiently large \(C\). We prove half of their prediction, showing that it is indeed impossible to cluster if \((a - b)^2 < 2(a + b)\). Furthermore we show that it is impossible even to estimate the model parameters from the graph when \((a - b)^2 < 2(a + b)\); on the other hand, we provide a simple and efficient algorithm for estimating \(a\) and \(b\) when \((a - b)^2 > 2(a + b)\). Following Decelle et al, our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems.
The clustering problem in its general form is, given a (possibly weighted) graph, to divide its vertices into several strongly connected classes with relatively weak cross-class connections. This problem is fundamental in modern statistics, machine learning and data mining, but its applications range from population genetics [29], where it is used to find genetically similar sub-populations, to image processing [33, 36], where it can be used to segment images or to group similar images, to the study of social networks [28], where it is used to find strongly connected groups of like-minded people.
Although sparse graphs are natural for modelling many large networks, the planted partition model seems to be most difficult to analyze in the sparse setting. Despite the large amount of work studying this model, the only results we know of that apply in the sparse case \(p, q = O(\frac1n)\) are those of Coja-Oghlan. Recently, Decelle et al. [9] made some fascinating conjectures for the cluster identification problem in the sparse planted partition model. In what follows, we will set \(p = a/n\) and \(q = b/n\) for some fixed \(a > b > 0\).
This second conjecture is based on a connection with the tree reconstruction problem (see [26] for a survey). Consider a multi-type branching process where there are two types of particles named \(+\) and \(-\). Each particle gives birth to \(\mathrmPois(a)\) (ie. a Poisson distribution with mean \(a\)) particles of the same type and \(\mathrmPois(b)\) particles of the complementary type. In the tree reconstruction problem, the goal is to recover the label of the root of the tree from the labels of level \(r\) where \(r \rightarrow \infty \). This problem goes back to Kesten and Stigum [23] in the 1960s, who showed that if \((a-b)^2 > 2(a+b)\) then it is possible to recover the root value with non-trivial probability. The converse was not resolved until 2000, when Evans et al. [12] proved that if \((a-b)^2 \le 2(a+b)\) then it is impossible to recover the root with probability bounded above \(1/2\) independent of \(r\). This is equivalent to the reconstruction or extremality threshold for the Ising model on a branching process.
At the intuitive level the connection between clustering and tree reconstruction, follows from the fact that the neighborhood of a vertex in \(\mathcal G(n, \fracan, \fracbn)\) should look like a random labelled tree with high probability. Moreover, the distribution of that labelled tree should converge as \(n \rightarrow \infty \) to the multi-type branching process defined above. We will make this connection formal later.
Decelle et al. also made a conjecture related to the the parameter estimation problem that was previously studied extensively in the statistics literature. Here the problem is to identify the parameters \(a\) and \(b\); note that if the blocks can be recovered exactly, then the parameters may easily be estimated simply by counting edges between different blocks; this was noted, for example, in [1]. Nevertheless, parameter estimation is interesting on its own, for instance because it is a first step for some clustering algorithms. Hence, [18] and [35] both discussed ML and Bayesian methods for parameter estimation, although neither work gave theoretical guarantees.
As in the clustering problem, Decelle et al. provided a parameter-estimation algorithm based on belief propagation and they used physical ideas to argue that there is a threshold above which the parameters can be estimated, and below which they cannot.
Theorem 1 is stronger than Conjecture 2 because it says that an even easier problem cannot be solved: if we take two random vertices of \(G\), Theorem 1 says that no algorithm can tell whether or not they have the same label. This is an easier task than finding a bisection, because finding a bisection is equivalent to labeling all the vertices; we are only asking whether two of them have the same label or not. Theorem 1 is also stronger than the conjecture because it includes the case \((a - b)^2 = 2(a+b)\), for which Decelle et al. did not conjecture any particular behavior.
To establish Theorem 3 we count the number of short cycles in \(G \sim \mathbb P_n\). It is well-known that the number of \(k\)-cycles in a graph drawn from \(\mathbb P'_n\) is approximately Poisson-distributed with mean \(\frac1k (\fraca+b2)^k\). The proof of this fact can be modified as in [4] to show a Poisson limit for cycle counts in more general inhomogeneous graphs. For completeness, we include the proof of our special case showing that the number of \(k\)-cycles in \(\mathbb P_n\) is approximately Poisson-distributed with mean \(\frac1k \big ((\fraca+b2)^k + (\fraca-b2)^k\big )\).
While there is in general no efficient algorithm for counting cycles in graphs, we show that with high probability the number of short cycles coincides with the number of non-backtracking walks of the same length which can be computed efficiently using matrix multiplication.
As mentioned earlier, Theorem 1 intuitively follows from the fact that the neighborhood of a vertex in \(\mathcal G(n, \fracan, \fracbn)\) should look like a random labelled tree with high probability and the distribution of that labelled tree should converge as \(n \rightarrow \infty \) to the multi-type branching process defined above. While this intuition is not too hard to justify for small neighborhoods (by proving there are no short cycles etc.) the global ramifications are more challenging to establish. This is because, conditioned on the graph structure, the model is neither an Ising model, nor a Markov random field! This is due to two effects:
This already shows that the second moment is bounded if \((a-b)^2 < 2(a+b)\). However, in order to establish the existence of a density, we also need to show that \(Y_n\) is bounded away from zero asymptotically. In order to establish this, we utilize the small graph conditioning method by calculating joint moments of the number of cycles and \(Y_n\). It is quite surprising that this calculation can be carried out in rather elegant manner, since many other applications of this method are much more technically involved.
The main result of this section is that the number of \(k\)-cycles of \(G \sim \mathbb P_n\) is approximately Poisson-distributed. We will then use this fact to show the first part of Theorem 2. The cycle counting result that we present is actually a special case of a result by Bollobs et al. [4], who show a Poisson limit for cycle counts in more general inhomogeneous graphs, which have continuous edge labels and a kernel that defines edge probabilities; our special case is recovered by taking the obvious two-valued kernel in [4, Theorem 17.1]. For completeness, we include the proof of our special case.
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