1 Introduction
Let us begin by recalling a quote of Henri Poincare:
Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house.
Cancer is driven by somatic mutations that target signaling and regulatory pathways that control cellular proliferation and cell death [25]. Understanding how this happens is of paramount importance in order to improve our ability to intervene and attack cancer. Since the advent of DNA sequencing technologies, our understanding has progressed enormously and resulted in useful therapies.^{5}^{5}5Current advances include the introduction of a single anticancer agents, which simply bind with growth factor receptors stopping abnormal cell proliferations. For instance, in the context of breast cancer, Herceptin antibody stops cells abnormal proliferation signals by binding with the excess of growth factor receptors on the cell surface caused by point mutations in the gene Her2. Gleevec is a second example. It is used to treat chronic myeloid leukemia, which is a type of blood cell tumor due to an inappropriate gene fusion product of a translocation in chromosomes 9 and 22. The resulting fusion gene BCRABL has an increased kinase activity, resulting in an increase in proliferation signals. Similar to Herceptin, Gleevec blocks the growth signals that the abnormal fusion gene generates and thus prevents the cell proliferation. More sophisticated approaches include immunotherapy that activates the immune system against cancer cells as well as the use of combinations of therapeutic agents to attack multiple pathways fundamental in cancer development, preventing resistance from occurring.
Notwithstanding the above, cancer morbidity is still very high and our understanding is still incomplete. Certainly, finding more relevant pathways will be helpful. Indeed we have developed novel formulations and algorithms that are computationally effective in doing so, a subject of our companion paper ([1] that builds on our work [2, 3]). We also believe that to start building a house it is not enough to simply accumulate more stones faster. To that end, in this paper, we look at whether the stones that we have identified so far form patterns that can help us build a house, using methods from algebraic topology (see Table 1).
The view we are taking here is also articulated in The Hallmarks of Cancer ([7]):
Two decades from now, having fully charted the wiring diagrams of every cellular signaling pathway, it will be possible to lay out the complete “integrated circuit of the cell” upon its current outline (Figure 1). We will then be able to apply the tools of mathematical modeling to explain how specific genetic lesions serve to reprogram this integrated circuit… One day, we imagine that cancer biology and treatment–at present, a patchwork quilt of cell biology, genetics, histopathology, biochemistry, immunology, and pharmacology–will become a science with a conceptual structure and logical coherence that rivals that of chemistry or physics.
Pathways  Algebraic topology 

Regulatory/signaling pathway  simplex 
The Emergent integrated circuit of the cell  simplicial complex 
Number of patients shared between two genes  persistent parameter 
Our topological journey departs from the (well trodden) path that recognized that signaling or regulatory pathways can be viewed as independent sets (modulo some notion of tolerance) in a matrix with rows as patients and columns as genes, which is used in determining new pathways via different computational methods (see our companion paper [1] and previous works [21, 22, 5, 18, 10]). Our key insight is that these pathways (however discovered), when grouped together, define a simplicial complex which is, pictorially, a polytope with faces given by those pathways. This vantage point connects us to the marvellous world of topology where simplicial complexes are the prototypes of spaces with shapes. Our journey explores this notion of shape in cancer genomics.
Our framework is built on this sequence of assignments:
MutationData  graphs  SpaceOfPathways  ShapeMeasurements  

where:

is the mutation graph. That is, is the graph defined by the set of genes in where two genes are connected if they have harboured mutations for the same patients. is the simplicial complex given by the set of all independent sets of . The underlying mathematics detailed in [6] is based on the socalled MayerVietoris construction, which itself is articulated around clique covering the graph . We prove in [6] that, for a large class of graphs, such coverings provide most of the simplices of .

is the space of pathways. The facets (maximal independent sets) of are the pathways, and their nerve defines the space of pathways . Pictorially, the space of pathways is visualized through its 1skeleton of the graph with pathways as vertices and where two pathways are connected by an edge if they intersect.

is the homology of the space of pathways, that is, the shape measurements of the space of pathways.
We have applied our approach to two different mutation data obtained from TCGA: Acute myeloid leukemia (AML) [14] and Glioblastoma multiforme (GBM) [13]. For both the data, we have computed the assignment
(1.1) 
using persistent homology.^{6}^{6}6In the language of Homotopy Type Theory (HoTT) [19], this procedure translates into: tumor
category. This category is an topos (a logical structure) in the sense of Lurie [12], so it obeys the axioms of HoTT.
Our calculation shows that the space of pathways for
AML mutation data is homotopy equivalent to a sphere, while in the case of GBM data, the space of pathways is homotopy equivalent to figure eight (genus2 surface).
2 Related works
The motivation for the present work originates from the work of the Raphael Lab, centred around the Dendrix algorithm [23], and its later improvements including CoMEt [9]. Both algorithms are in widespread use in wholegenome analysis–for instance, in [16, 17, 4, 20]. Building on those foundations, our work extends in the following two directions: First, the two key notions of exclusivity and coverage are abstracted here by the two simplicial complexes and or, more precisely, by the filtrations of simplicial complexes and , where we allow the two parameters to vary: and . We do so because, in reality, the assumptions that driver pathways exhibit both high coverage and high exclusivity need not to be strictly satisfied. The functoriality of persistent homology, which takes the two filterations as input, handles this elegantly; we decide on the values of these two parameters after computing the barcodes. This type of analysis is not present in the aforementioned works. Second, motivated by the naturality of the constructions, the present paper goes beyond the computational aspects and ventures into the conceptuality of cancer. We have introduced the notion of the topological space of pathways , together with its homology spaces, as a paradigm to rationalize the extraordinary complexity of cancer. To the best of our knowledge, this is a “provocative” idea that has not been explored before.
Another related set of works is [11] and [15] (from Stanford’s applied topology group), which also use algebraic topology tools in cancer. However, they differ from ours on two counts: the nature of the problem treated and the methodology used. The algorithm introduced, called Mapper, is a topological clustering algorithm, and it is not based on persistent homology. Mapper was used to identify a subgroup of breast cancers with excellent survival, solely based on topological properties of the data.^{7}^{7}7Commercial applications of this approach in various different areas of practice are tackled through Ayasdi.
3 Connecting to algebraic topology
Different errors occurring during data preparation (i.e., sequencing step, etc.) affect the robustness of the results. This implies that the computed pathways are likely to be affected by these errors and can not be considered as a robust finding without explicitly modeling the error in our constructions. To that end, the assignment we mentioned above is done as follows:

We think about the number of patients that are shared between two genes as a parameter (thus, absolute exclusivity corresponds to taking this parameter to zero).

Instead of applying our procedure once, we apply it for a range of values of the exclusivity parameter . That is, we consider a filtration of graphs (instead of one):
(3.1) where for each graph two genes are connected if they have harboured mutations concurrently for at least patients. This yields a second filtration of simplicial complexes (we call such a filtration, a persistent pathway complex)
(3.2) where is one of the three complexes we define below.

Measure the shape (the homology) of the different pathway spaces and then “average” the shape measurements that are obtained.
This (practical) version of homology is what we refer to as persistent homology. It tracks the persistent topological features through a range of values of the parameter; genuine topological properties persist through the change of the parameter whilst noisy observations do not (all of these will be made precise below). The key point is that the mapping is functorial; that is, it sends a whole filtration (i.e., 3.1) into another filtration (i.e., 3.2). In other words, it is not only sending graphs to simplicial complexes but it is also preserving their relations. This functoriality is at the heart of persistent homology.
3.1 The GenePatient graph
Consider a mutation data for tumors (i.e., patients), where each of the genes is tested for a somatic mutation in each patient. To this data we associate a mutation matrix with rows and columns, where each row represents a patient and each column represents a gene. The entry in row and column is equal to 1 if patient harbours a mutation in gene and it is 0 otherwise. For a gene , we define the fiber
(3.3) 
Definition 1
The mutation graph associated to and is the graph whose vertex set is the set of genes and whose edges are pairs of genes such that
(3.4) 
3.2 The space of pathways
We would like to assign to the mutation graph an independence complex. We present below two different functorial ways to do so.
Definition 2
Given a mutation graph , its persistent pathway complex is the independence complex of (or equivalently, the clique complex of , the complement of ).
We also define the persistent pathway complex .
Definition 3
The persistent pathway complex is defined as follows. Fix and let . The complex is the complex generated by all independent sets of with coverage
(3.5) 
where the counting is done without redundancy.
The following definition is also valid for the persistent pathway complex .
Definition 4
The space of pathways of a persistent pathway complex is the nerve generated by the facets of the complex, that is, the simplicial complex where is a simplex if and only if the facets indexed with have a non empty intersection. We denote the space of pathways by .
The space of pathways is visualized through its 1skeleton: the graph with pathways as vertices, and two pathways are connected if they intersect (see Figures 3 and 4).
3.3 A primer on persistent homology
Recall that our plan is to compute the persistent homology of the independence complex . We have explained that this is simply computing the homology of for a range of increasing values of the parameter . In this section, we explain this notion of homology (which we have introduced as measurements of the shape of the space ). For simplicity, we drop out the subscript from the complex . It is also more convenient to introduce homology for clique complexes (for independence complexes, it suffices to replace, everywhere below, cliques with independent sets).
The homology of the simplicial complex (now a clique complex of ) is a sequence of
vector spaces (i.e., vector spaces with integer coefficients):
(3.6) 
defined as follows:

The zeroth vector space is spanned by all connected components of ; thus, the dimension gives the number of connected components of the space.

The first homology space is spanned by all closed chains of edges (cycles) in which are not triangles – see Figure 3; in this case, the dimension gives the number of “holes” in the space.

Similarly, the second space is spanned by all 2dimensional enclosed three dimensional “voids” that are not tetrahedra (as in Figure 3 below).
Higher dimensional spaces are defined in a similar way (although less visual). Their dimensions count non trivial high dimensional voids. The dimensions are called Betti numbers and provide the formal description of the concept of shape measurements.^{8}^{8}8In textbooks, one typically starts with a continuous space (for instance, a doughnut shaped surface) and then triangulizes it, yielding the simplicial complex . The homology of the continuous space is the homology of its discretization that we introduced pictorially above. Appendix A, on page A, explains how to compute homology using quantum computers or QUBOsolvers such as [2].
We move now to the notion of persistent homology and make it a bit more precise. For that, let us reintroduce the persistent parameter and let be again an independence complex. It is clear that if and then the pair , which is an edge in is also is an edge in . This means that is a subgraph of ; thus, we have whenever (since an independent set for a given graph is also an independent set for any of its subgraphs). The mapping is functorial. It turns out that homology itself is functorial and all this functoriality is the mathematical reason that the following is correct: one can track the Betti numbers over a range of values and consider the subrange where the Betti numbers are not changing (significantly). Pathways within this subrange are considered to have passed our test and declared robust computations.
4 Real mutation data
We have applied our approach to two mutation data (formulations and algorithms are available in [1]): Acute myeloid leukemia [14] and Glioblastoma multiforme [13]. For both data, we have computed the assignment tumor pathways through persistent pathway complexes (thus, declared robust output). The complete result is presented in long tables given in the Appendix. Interestingly, our calculation also shows that AML data is homotopy equivalent to a sphere while GBM data is homotopy equivalent to figure eight (genus2 surface).
4.1 Acute myeloid leukemia data
The data has a cohort of 200 patients and 33 genes ([14]). We have chosen the coverage threshold patient. We also neglected all genes that have fewer than 6 patients. These numbers are chosen based on the stability of barecodes for pairs , while barecodes for pairs less than (6, 80) exhibit strong variations. This is also consistent with the fact that choosing genes with fewer than 5 or 6 patients is not common in such studies (genes with low numbers of patients are not considered robust enough, and are very prone to errors. This extra precaution is commonly used in the field). Now for the numbers of patients and coverage we have chosen, the Betti numbers are computed for various values of in the table below:
1  6  0.86  
2  84  0.97  
3  50  1 
Figure 4 below gives the 1skeleton of the nerve for . The Betti numbers are not changing; thus, is a reasonable choice. Recall that each node represents a pathway and two pathways are connected if they intersect (as sets of genes). We have used different colors to represent different pathways as described in the table below (no other meaning for the coloring).
Color  Genes in the pathway 

Blue  ’PML.RARA’, ’MYH11.CBFB’, ’RUNX1.RUNX1T1’, ’TP53’, ’NPM1’, ’RUNX1’ 
Blue light  ’PML.RARA’, ’MYH11.CBFB’, ’RUNX1.RUNX1T1’, ’TP53’, ’NPM1’, ’MLL.PTD’ 
Orange  ’PML.RARA’, ’MYH11.CBFB’, ’RUNX1.RUNX1T1’, ’DNMT3A’ 
Orange light  ’Other Tyr kinases’, ’MYH11.CBFB’, ’MLL.PTD’, ’NPM1’ 
Green  ’MLLX fusions’, ’TP53’, ’FLT3’ 
Green light  ’Other Tyr kinases’, ’MYH11.CBFB’, ’DNMT3A’, ’MLLX fusions’ 
4.2 Glioblastoma multiforme data
The second mutation data is taken from [13]. It has 84 patients and around 100 genes. Approximately, 70% of the genes have very low coverage so we removed them from the data; precisely, we have removed all genes with fewer than 10 patients. We have used the complex , with fixed to 7 because lower values don’t exhibit stable topologies; that is, barcodes are essentially one point long. This is a different choice of complex from the complex that we used with the previous AML data, which interestingly doesn’t yield noticeable stability–a possible explanation of this is the small size of the data (i.e., number of patients). Recall that the definitions of the two complexes are given on page 3.2 (definitions 2 and 3). In the table below, one can see that the topology stabilizes for the first three values of . Any choice of pathways within this range is considered robust (proportionate to the small size of the data).
66  15  0.73  
67  14  0.74  
68  12  0.72  
69  6  0.6  
70  50  1 
The following table provides the legend for the Figure 5 corresponding to the GBM data:
Color  Genes in the pathway 

blue  RB1, NF1, CYP27B1, CDKN2B 
blue light  RB1, NF1, MDM2, AVILCTDSP2, CDKN2B 
orange  TP53, MDM2, OS9, CDKN2A 
orange light  TP53, MDM2, AVILCTDSP2, CDKN2A 
green  TP53, MDM2, DTX3, CDKN2A 
green light  RB1, NF1, CDK4, CDKN2B 
brown  TP53, CDK4, CDKN2A 
brown light  TP53, CYP27B1, CDKN2A 
purple  TP53, MDM2, AVILCTDSP2, MTAP 
purple light  TP53, MDM2, DTX3, MTAP 
red  RB1, NF1, MDM2, OS9, CDKN2B 
pink  TP53, MDM2, OS9, MTAP 
5 Conclusion
The main goal of this paper was to suggest a study of the space of cancer pathways, using the natural language of algebraic topology. We hope that the consideration of the pathways collectively, that is, as a topological space, helps to reveal novel relations between these pathways. Indeed, we have seen that the homology in the case of AML indicates that the mutation data has the shape of a sphere.^{9}^{9}9Using a different visual representation, Vogelstein found AML to be very different from other cancers. Indeed, that has allowed many scientists to speculate that such genetically simple tumors are more susceptible to drugs, and thus intrinsically more curable.
However, in the case of GBM, the final set of pathways has the topology of a double torus (or, more technically, a genus2 surface). This intriguing observation raises the question of whether these facts translate into a new biological understanding about cancer. Studying the space of pathways of other cancers will be illuminating as well, if they also show similar structures, and we can classify cancers by the topology of their mutated driver pathways. This is an example of the new type of hypotheses one can now formulate about the data. Eventually, our goal (recalling Poincare) is to help build a house by revealing patterns among the stones. Let us close with a quote from
The Emperor of All Maladies (page 458):The third,^{10}^{10}10The first is targeted therapy on the mutated pathways, as we mentioned in the Introduction. The second is cancer prevention through identifying preventable carcinogens. and arguably most complex, new direction for cancer medicine is to integrate our understanding of aberrant genes and pathways to explain the behavior of cancer as a whole, thereby renewing the cycle of knowledge, discovery and therapeutic intervention.
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Appendix A Appendix
This appendix has three parts. In the first part, we review an efficient procedure for computing homology groups. The remaining two parts give the obtained list of pathways for GBM and AML data, respectively.
a.1 Computing homology on quantum computers
We provide the following material (from [6]) for easy access.
We review how the homology spaces are computed. In principle, the formal definitions of homology that can be found in any algebraic topology textbook, are sufficient for computations. However, we point here to a more efficient approach based on MayerVietoris blowup complexes. We
formulate finding optimal MayerVietoris blowup complexes as Quadratic Unconstrained Binary Optimizations (QUBO). QUBOsolvers (such as the DWave quantum annealer [8] or quantuminspired classical solvers such as [2]) are now available for calculations.
Let be a cover of by simplicial subcomplexes ; here again we have dropped the subscript from for simplicity. For we define
Definition 5
The MayerVietoris blowup complex of the simplicial complex and cover is defined by:
A basis for the chains is . The boundary of a cell is given by:
Simply put, the simplicial complex is the set of the “original” simplices in addition to the ones we get by blowing up common simplices. These are of the form in the definition above. In Figure 6, the yellow vertex common to the two subcomplexes is blownup into an edge , and the edge is blownup into the “triangle” . In Figure 7, the vertex common to three subcomplexes is blownup into the triangle .
We will not prove it here, but the projection is a homotopy equivalence and induces an isomorphism [27]. The key point is that the boundary map of the simplicial complex (which replaces by the homotopy equivalence) has a nice block form suitable for parallel rank computation. As an example, let us consider again the simplicial complex depicted in Figure 6. First, , the space of vertices of the blowup complex , is spanned by the vertices
that is, all vertices of taking into account the partition they belong to. The space of edges is spanned by
which is the set of the “original” edges (edges of the form with and is an edge in ) and the new ones resulting from blowups, that is, those of the form where is a vertex in (if the intersection is empty, the value of boundary map is just 0). The matrix of the boundary map with respect to the given ordering is then:
Clearly, one can now rowreduce each coloured block independently. There might be remainders, that is, zero rows except for the intersection part. We collect all such rows in one extra matrix and rowreduce it at the end and aggregate. For the second boundary matrix we need to determine . The 2simplices are of three forms. First, the original ones (those of the form with ; in this example there is none) then those of the form , with being in . And finally, those of the form , with (there is none in this example, but Figure 5 has one). We get ; thus, there is no need for parallel computation.
Finally, for efficient computations, it is necessary that the homologies of the smaller blocks are easy to compute. This is where the QUBOsolver comes in. It is used to obtain a “good” cover by computing a clique cover of the original graph (plus a completion step). It is easy to see that such coverings do, indeed, come with trivial block homologies [6].
a.2 GBM pathways list
The table below reports the final pathways in Figure 5. Details on the data, the parameters, and how the pathways are computed using persistent homology, can be found in Subsection 4.2 on page 4.2.
Pathway  coverage 

RB1 , NF1 , CYP27B1 , CDKN2B  70 
RB1 , NF1 , MDM2 , AVILCTDSP2 , CDKN2B  69 
TP53 , MDM2 , OS9 , CDKN2A  69 
TP53 , MDM2 , AVILCTDSP2 , CDKN2A  69 
TP53 , MDM2 , DTX3 , CDKN2A  69 
RB1 , NF1 , CDK4 , CDKN2B  69 
RB1 , NF1 , MDM2 , OS9 , CDKN2B  68 
TP53 , MDM2 , OS9 , MTAP  68 
TP53 , MDM2 , AVILCTDSP2 , MTAP  68 
TP53 , MDM2 , DTX3 , MTAP  68 
TP53 , CDK4 , CDKN2A  68 
TP53 , CYP27B1 , CDKN2A  68 
TP53 , CDK4 , MTAP  67 
TP53 , CYP27B1 , MTAP  67 
RB1 , NF1 , CYP27B1 , CDKN2A  66 
RB1 , NF1 , MDM2 , OS9 , CDKN2A  65 
RB1 , NF1 , MDM2 , AVILCTDSP2 , CDKN2A  65 
RB1 , NF1 , MDM2 , DTX3 , CDKN2B  65 
RB1 , NF1 , CDK4 , CDKN2A  65 
RB1 , NF1 , CYP27B1 , MTAP  65 
RB1 , NF1 , MDM2 , OS9 , MTAP  64 
RB1 , NF1 , MDM2 , AVILCTDSP2 , MTAP  64 
RB1 , NF1 , CDK4 , MTAP  64 
RB1 , NF1 , MDM2 , DTX3 , CDKN2A  62 
RB1 , NF1 , MDM2 , DTX3 , MTAP  61 
TP53 , MDM2 , AVILCTDSP2 , SEC61G  60 
RB1 , NF1 , EGFR , OS9  60 
TP53 , MDM2 , OS9 , SEC61G  59 
TP53 , MDM2 , DTX3 , SEC61G  59 
PTEN , IFNA21 , MDM2 , AVILCTDSP2  58 
TP53 , MDM2 , AVILCTDSP2 , IFNA21  58 
RB1 , NF1 , EGFR , DTX3  58 
PTEN , IFNA21 , CYP27B1  58 
PTEN , IFNA21 , MDM2 , OS9  57 
TP53 , MDM2 , OS9 , IFNA21  57 
TP53 , MDM2 , DTX3 , IFNA21  57 
PTEN , IFNA21 , CDK4  57 
RB1 , NF1 , MDM2 , AVILCTDSP2 , SEC61G  55 
PTEN , IFNA21 , MDM2 , DTX3  55 
TP53 , MDM2 , AVILCTDSP2 , ELAVL2  55 
RB1 , NF1 , MDM2 , OS9 , SEC61G  54 
TP53 , MDM2 , OS9 , ELAVL2  54 
TP53 , MDM2 , DTX3 , ELAVL2  54 
RB1 , NF1 , MDM2 , AVILCTDSP2 , IFNA21  53 
TP53 , CDK4 , IFNA21  53 
TP53 , CDK4 , ELAVL2  53 
TP53 , CYP27B1 , IFNA21  53 
TP53 , CYP27B1 , ELAVL2  53 
RB1 , NF1 , MDM2 , OS9 , IFNA21  52 
RB1 , NF1 , MDM2 , DTX3 , SEC61G  52 
RB1 , NF1 , MDM2 , AVILCTDSP2 , ELAVL2  51 
RB1 , NF1 , CYP27B1 , IFNA21  51 
RB1 , NF1 , CYP27B1 , ELAVL2  51 
RB1 , NF1 , MDM2 , OS9 , ELAVL2  50 
RB1 , NF1 , CDK4 , IFNA21  50 
RB1 , NF1 , CDK4 , ELAVL2  50 
RB1 , NF1 , MDM2 , DTX3 , IFNA21  49 
RB1 , NF1 , MDM2 , DTX3 , ELAVL2  47 
a.3 AML pathways list
The following table gives the list of all 65 pathways of (Figure 4) and their coverage. Details about the data and how the pathways are obtained using persistent homology can be found in Section 3.1. All pathways listed below passed our robustness test.
Pathway  Coverage 

PML.RARA , MYH11.CBFB , RUNX1.RUNX1T1 , TP53 , NPM1 , RUNX1  123 
PML.RARA , MYH11.CBFB , RUNX1.RUNX1T1 , TP53 , NPM1 , MLL.PTD  113 
PML.RARA , MYH11.CBFB , RUNX1.RUNX1T1 , DNMT3A  85 
Other Tyr kinases , MYH11.CBFB , MLL.PTD , NPM1  83 
MLLX fusions , TP53 , FLT3  81 
Other Tyr kinases , MYH11.CBFB , DNMT3A , MLLX fusions  80 
PML.RARA , MYH11.CBFB , Cohesin , Other modifiers  78 
MLLX fusions , DNMT3A , RUNX1.RUNX1T1 , MYH11.CBFB  78 
PML.RARA , MYH11.CBFB , RUNX1.RUNX1T1 , TP53 , PHF6 , IDH1  75 
Other myeloid TFs , MYH11.CBFB , Cohesin , Other modifiers  74 
Other Tyr kinases , FLT3 , MLLX fusions  74 
PML.RARA , MYH11.CBFB , RUNX1.RUNX1T1 , TP53 , IDH2  70 
PML.RARA , MYH11.CBFB , RUNX1.RUNX1T1 , CEBPA , RUNX1  66 
PML.RARA , MYH11.CBFB , RUNX1.RUNX1T1 , TP53 , PHF6 , MLL.PTD  65 
Other myeloid TFs , MYH11.CBFB , TP53 , RUNX1.RUNX1T1 , RUNX1  65 
PML.RARA , PTPs , IDH2 , TET2  65 
PML.RARA , MYH11.CBFB , TET2 , IDH2  64 
MLLX fusions , TP53 , RUNX1.RUNX1T1 , MYH11.CBFB , IDH2  63 
PML.RARA , MYH11.CBFB , RUNX1.RUNX1T1 , CEBPA , PHF6 , MLL.PTD  62 
PML.RARA , KRAS/NRAS , PHF6 , MLL.PTD , KIT  62 
MLLX fusions , TP53 , RUNX1.RUNX1T1 , MYH11.CBFB , IDH1  62 
MLLX fusions , TP53 , RUNX1.RUNX1T1 , MYH11.CBFB , RUNX1  62 
Other myeloid TFs , MYH11.CBFB , TP53 , RUNX1.RUNX1T1 , PHF6 , MLL.PTD  61 
PML.RARA , KRAS/NRAS , PHF6 , MLL.PTD , RUNX1.RUNX1T1  61 
PML.RARA , KIT , TP53 , IDH2  60 
PML.RARA , KIT , TP53 , RUNX1  59 
MLLX fusions , TET2 , MYH11.CBFB , IDH2  57 
PML.RARA , PTPs , IDH2 , KIT  56 
PML.RARA , KIT , CEBPA , RUNX1  56 
PML.RARA , KIT , TP53 , PHF6 , MLL.PTD  55 
PML.RARA , PTPs , IDH2 , RUNX1.RUNX1T1  55 
PML.RARA , PTPs , RUNX1 , KIT  55 
Other myeloid TFs , KIT , TP53 , RUNX1  55 
PML.RARA , PTPs , RUNX1 , RUNX1.RUNX1T1  54 
MLLX fusions , TP53 , KIT , IDH2  53 
PML.RARA , KIT , CEBPA , PHF6 , MLL.PTD  52 
MLLX fusions , TP53 , RUNX1.RUNX1T1 , MYH11.CBFB , MLL.PTD  52 
SerTyr kinases , RUNX1.RUNX1T1 , PTPs , MLL.PTD  52 
MLLX fusions , TP53 , KIT , RUNX1  52 
Pathway  Coverage 

Other myeloid TFs , KIT , TP53 , PHF6 , MLL.PTD  51 
PML.RARA , WT1 , RUNX1.RUNX1T1 , TP53  51 
PML.RARA , MYH11.CBFB , TET2 , PHF6  50 
PML.RARA , KIT , Cohesin  50 
Other Tyr kinases , MYH11.CBFB , IDH2 , MLLX fusions  49 
Other Tyr kinases , MYH11.CBFB , IDH1 , MLLX fusions  48 
Other Tyr kinases , MYH11.CBFB , MLL.PTD , PHF6 , Other myeloid TFs  47 
Other Tyr kinases , KRAS/NRAS , PHF6 , MLL.PTD  47 
Other myeloid TFs , MYH11.CBFB , TET2 , PHF6  46 
MLLX fusions , Cohesin , MYH11.CBFB  46 
Other myeloid TFs , KIT , Cohesin  46 
Other Tyr kinases , MYH11.CBFB , IDH1 , PHF6  45 
PML.RARA , PTPs , MLL.PTD , KIT  45 
PML.RARA , WT1 , TET2  45 
PML.RARA , PTPs , MLL.PTD , RUNX1.RUNX1T1  44 
MLLX fusions , TP53 , RUNX1.RUNX1T1 , WT1  44 
MLLX fusions , Cohesin , KIT  43 
MLLX fusions , TP53 , KIT , MLL.PTD  42 
Other Tyr kinases , PTPs , IDH2  41 
Other Tyr kinases , MYH11.CBFB , MLL.PTD , MLLX fusions  38 
MLLX fusions , TET2 , WT1  38 
Spliceosome , CEBPA  37 
Spliceosome , PTPs  36 
Spliceosome , Other myeloid TFs  36 
Other Tyr kinases , WT1 , MLLX fusions  30 
Other Tyr kinases , PTPs , MLL.PTD  30 
The following long table gives the assignment tumor list of pathways for the AML data. For each tumor we assign a robust (in our topological sense) set of pathways. More specifically, for a tumor (sample ID) i, the second column gives the list of all pathways that have passed our robustness test.
Sample ID  Pathways  Cyto 

0  0, 1, 2, 6, 8, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
1  0, 1, 2, 7, 8, 11, 12, 13, 14, 17, 18, 20, 21, 22, 23, 30, 33, 36, 37, 40, 53, 54  Good 
2  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 26, 36, 41, 43, 44, 45, 46, 47, 48, 50, 58  Good 
3  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 26, 36, 41, 43, 44, 45, 47, 48, 50, 58  Good 
4  0, 1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 43, 44, 45, 46, 50, 51, 52, 53, 57, 58, 63, 64  Good 
5  0, 1, 2, 4, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
6  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 26, 27, 30, 31, 33, 36, 37, 41, 43, 44, 45, 47, 48, 50, 51, 53, 57, 58, 61, 64  Good 
7  0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45,  
46, 49, 50, 51, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64  Good  
8  0, 1, 2, 4, 7, 8, 10, 11, 12, 13, 14, 17, 18, 20, 21, 22, 23, 30, 33, 36, 37, 40, 53, 54  Good 
9  0, 1, 2, 6, 8, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
10  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 26, 36, 41, 43, 44, 45, 47, 48, 50, 58  Good 
11  0, 1, 2, 6, 8, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53, 60, 61, 62  Good 
12  40, 52, 54, 59, 60, 61, 62, 63  Good 
13  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49,  
50, 51, 55, 56, 58  Good  
14  0, 1, 2, 4, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
15  0, 1, 2, 6, 8, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
16  0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 20, 21, 22, 23, 30, 33, 36, 37, 40, 42, 43, 44, 45, 46, 48, 49, 50, 53, 54, 55, 57, 58, 63, 64  Good 
17  0, 1, 2, 4, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
18  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48,  
49, 50, 51, 55, 56,  
58, 60, 61, 62  Good  
19  0, 1, 2, 6, 8, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
20  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 26, 36, 41, 43, 44, 45, 47, 48, 50, 58  Good 
21  0, 1, 2, 7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 47, 49,  
51, 52, 53, 54, 55, 56, 59, 60, 61, 62  Good  
22  32, 4, 39, 9, 10, 45, 14, 47, 49, 22, 62  Good 
23  0, 1, 2, 6, 8, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
24  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 26, 36, 41, 43, 44, 45, 47, 48, 50, 58  Good 
25  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48,  
49, 50, 51, 52, 54, 55, 56, 58, 59, 63  Good  
26  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 26, 36, 41, 43, 44, 45, 46, 47, 48, 50, 58  Good 
27  0, 1, 2, 6, 8, 9, 11, 12, 13, 14, 15, 16, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 39, 40, 41, 42, 45, 47, 49, 51, 52, 53, 62  Good 
28  0, 1, 2, 4, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
29  0, 1, 2, 6, 7, 8, 9, 11, 12, 13, 14, 17, 18, 20, 21, 22, 23, 30, 33, 36, 37, 40, 42, 48, 49, 53, 54, 55  Good 
30  0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 38, 40, 41, 42, 43, 44, 48,  
51, 52, 53, 54, 55, 56, 58, 59, 63  Good  
31  37  Good 
32  0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 20, 21, 22, 26, 36, 41, 43, 44, 45, 47, 48, 50, 58  Good 
33  0, 1, 2, 3, 5, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 43, 44, 45, 46, 50, 51, 52, 53, 57, 58, 63, 64  Good 
34  0, 1, 2, 6, 7, 8, 9, 11, 12, 13, 14, 17, 18, 20, 21, 22, 23, 30, 33, 36, 37, 40, 42, 48, 49, 53, 54, 55  Good 
35  0, 1, 2, 4, 6, 8, 10, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 40, 41, 42, 51, 52, 53  Good 
36  10, 4  Good 
37  0, 1, 2, 3, 5, 7, 8, 64, 44, 61, 15, 50, 51, 20, 53, 57, 33, 27, 37, 30, 31  Intermediate 
38  35, 6, 39, 8, 41, 13, 46, 45, 50, 18, 19, 9, 22, 23, 47, 29  Intermediate 
39  0, 32, 34, 43, 38, 33, 11, 12, 14, 15, 16, 17, 21, 25, 24, 57, 26, 27, 28, 30, 31  Intermediate 
40  0, 11, 12, 14, 15, 16, 17, 21, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 37, 38, 43, 57  Intermediate 
41  35, 60, 12, 46, 18, 19, 23, 28  Intermediate 
42  0, 1, 2, 3, 5, 7  Intermediate 
43  0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 48, 49, 55, 42  Intermediate 
44  0, 6, 9, 12, 14, 15, 16, 19, 21, 23, 25, 26, 28, 31, 32, 33, 38, 41, 42, 46, 47, 48, 49, 52, 55, 59, 60, 61, 62  Intermediate 
45  0, 1, 10, 3, 4  Intermediate 
46  2, 43, 5, 7, 11, 34, 46, 15, 16, 17, 19, 23, 24, 57, 26, 27, 30, 37  Intermediate 
47  10, 4  Intermediate 
48  0, 1, 3, 4, 6, 9, 10, 15, 27, 30, 31, 33, 37, 42, 48, 49, 51, 53, 55, 57, 61, 64  Intermediate 
49  0, 1, 2, 3, 4, 5, 7, 10  Intermediate 
50  0, 1, 2, 3, 4, 5, 7, 41, 10, 15, 16, 59, 52, 26, 47  Intermediate 
51  0, 1, 3, 4, 6, 9, 10, 15, 27, 30, 31, 33, 37, 42, 48, 49, 51, 53, 55, 57, 61, 64  Intermediate 
52  0, 1, 3, 37, 6, 33, 64, 9, 42, 15, 48, 49, 51, 53, 55, 57, 27, 61, 30, 31  Intermediate 
53  2, 59, 5, 7, 41, 15, 16, 52, 26, 47  Intermediate 
54  0, 1, 2, 3, 4, 5, 7, 10, 46, 19, 23  Intermediate 
55  0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 48, 49, 55, 42  Intermediate 
56  0, 1, 2, 3, 4, 5, 7, 10  Intermediate 
57  1, 3, 8, 13, 18, 19, 20, 22, 23, 29, 35, 36, 37, 39, 44, 45, 46, 50, 51, 53, 56, 58, 64  Intermediate 
58  0, 2, 5, 6, 7, 8, 9, 12, 13, 14, 18, 19, 21, 22, 23, 25, 28, 29, 31, 32, 33, 35, 37, 38, 39, 41, 42, 45, 46, 47, 48, 49, 50, 55, 60, 61, 62  Intermediate 
59  2, 5, 7, 46, 19, 23, 60, 61, 62  Intermediate 
60  0, 1, 3, 4, 6, 9, 10, 48, 49, 55, 42  Intermediate 
61  0, 1, 3, 4, 6, 40, 9, 10, 48, 49, 52, 54, 55, 59, 42, 63  Intermediate 
62  10, 4  Intermediate 
63  28, 18, 35, 12, 60  Intermediate 
64  0, 8, 11, 12, 14, 15, 16, 17, 20, 21, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 38, 43, 44, 50, 57, 60, 61, 62  Intermediate 
65  48, 2, 59, 5, 6, 7, 8, 41, 42, 55, 44, 15, 16, 49, 50, 52, 9, 20, 26, 47  Intermediate 
66  32, 2, 5, 7, 9, 39, 45, 14, 47, 49, 22, 62  Intermediate 
67  0, 1, 2, 3, 5, 6, 7, 9, 42, 48, 49, 55  Intermediate 
68  0, 1, 2, 3, 4, 5, 7, 10  Intermediate 
69  0, 1, 2, 3, 5, 6, 7, 40, 9, 44, 8, 50, 52, 54, 20, 59, 63  Intermediate 
70  0, 1, 34, 3, 11, 46, 15, 16, 17, 19, 43, 23, 24, 57, 26, 27, 30  Intermediate 
71  32, 39, 9, 45, 14, 47, 49, 22, 62  Intermediate 
72  0, 1, 34, 3, 4, 10, 11, 15, 16, 17, 43, 24, 57, 26, 27, 30  Intermediate 
73  0, 1, 3, 8, 44, 50, 20  Intermediate 
74  0, 1, 3, 4, 41, 10, 15, 16, 59, 52, 26, 47  Intermediate 
75  35, 4, 60, 10, 12, 18, 28  Intermediate 
76  0, 1, 2, 3, 4, 5, 7, 10  Intermediate 
77  1, 2, 3, 4, 5, 6, 7, 9, 10, 13, 15, 16, 18, 19, 22, 23, 26, 29, 35, 36, 37, 39, 41, 45, 46, 47, 51, 52, 53, 56, 58, 59, 64  Intermediate 
78  10, 4  Intermediate 
79  2, 5, 7  Intermediate 
80  0, 1, 2, 3, 4, 5, 7, 9, 10, 39, 45, 14, 47, 49, 32, 22, 62  Intermediate 
81  34, 4, 5, 38, 7, 10, 43, 44, 48, 17, 20, 21, 54, 55, 56, 36, 26, 59, 58, 63  Intermediate 
82  4, 6, 8, 9, 10, 44, 50, 20  Intermediate 
83  2, 5, 7, 9, 14, 15, 22, 27, 30, 31, 32, 33, 37, 39, 45, 47, 49, 51, 53, 57, 61, 62, 64  Intermediate 
84  0, 1, 3, 4, 10, 12, 13, 14, 18, 19, 21, 22, 23, 25, 28, 29, 31, 32, 33, 35, 36, 37, 38, 39, 40, 45, 46, 51, 52, 53, 54, 56, 58, 59, 63, 64  Intermediate 

85  9, 10, 4, 6  Intermediate 
86  0, 1, 3  Intermediate 
87  0, 1, 10, 3, 4  Intermediate 
88  34, 43, 37, 11, 15, 16, 17, 62, 24, 57, 26, 27, 60, 61, 30  Intermediate 
89  0, 1, 3  Intermediate 
90  0, 1, 2, 3, 5, 7, 8, 64, 44, 61, 15, 50, 51, 20, 53, 57, 33, 27, 37, 30, 31  Intermediate 
91  0, 1, 2, 3, 5, 6, 7, 9, 12, 15, 18, 19, 23, 27, 28, 30, 31, 33, 35, 37, 40, 42, 46, 48, 49, 51, 52, 53, 54, 55, 57, 59, 60, 61, 63, 64  Intermediate 
92  0, 32, 34, 43, 38, 33, 11, 12, 14, 15, 16, 17, 21, 25, 24, 57, 26, 27, 28, 30, 31  Intermediate 
93  59, 37, 41, 47, 16, 52, 26, 15, 60, 61, 62  Intermediate 
94  0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 39, 44, 45, 14, 47, 49, 50, 20, 32, 22, 62  Intermediate 
95  0, 1, 2, 3, 5, 6, 7, 9, 42, 55, 46, 48, 49, 19, 23  Intermediate 
96  0, 32, 38, 33, 12, 14, 21, 25, 28, 31  Intermediate 
97  59, 41, 46, 47, 16, 19, 52, 23, 26, 15  Intermediate 
98  0, 1, 10, 3, 4  Intermediate 
99  0, 1, 3, 37, 8, 44, 50, 20  Intermediate 
100  0, 1, 2, 4, 5, 7, 8, 11, 13, 14, 17, 20, 21, 22, 24, 25, 29, 32, 34, 36, 37, 38, 39, 40, 54, 56  Intermediate 
101  28, 18, 35, 12, 60  Intermediate 
102  0, 1, 3, 8, 44, 50, 20  Intermediate 
103  0, 32, 38, 33, 12, 14, 21, 25, 28, 31  Intermediate 
104  34, 4, 5, 38, 7, 10, 43, 44, 48, 17, 20, 21, 54, 55, 56, 36, 26, 59, 58, 63  Intermediate 
105  32, 51, 34, 35, 38, 39, 60, 42, 61, 49, 19, 56, 55, 24, 25, 27, 28, 29, 62, 31  Intermediate 
106  0, 1, 34, 3, 37, 11, 15, 16, 17, 43, 24, 57, 26, 27, 30  Intermediate 
107  0, 3, 5, 10, 12, 14, 21, 25, 28, 31, 32, 33, 38, 43, 44, 45, 46, 50, 57, 58, 63, 64  Intermediate 
108  4, 5, 6, 7, 9, 10, 12, 17, 18, 20, 21, 26, 28, 34, 35, 36, 38, 43, 44, 48, 54, 55, 56, 58, 59, 60, 63  Intermediate 
109  3, 5, 10, 12, 15, 16, 18, 26, 28, 35, 41, 43, 44, 45, 46, 47, 50, 52, 57, 58, 59, 60, 63, 64  Intermediate 
110  4, 40, 10, 52, 54, 59, 63  Intermediate 
111  0, 1, 2, 3, 4, 5, 7, 10  Intermediate 
112  0, 1, 3, 4, 5, 6, 7, 9, 10, 17, 19, 20, 21, 23, 26, 34, 36, 38, 43, 44, 46, 48, 54, 55, 56, 58, 59, 63  Intermediate 
113  0, 1, 3  Intermediate 
114  0, 1, 3, 12, 13, 14, 18, 19, 21, 22, 23, 25, 28, 29, 31, 32, 33, 35, 36, 37, 38, 39, 45, 46, 51, 53, 56, 58, 60, 61, 62, 64  Intermediate 
115  0, 1, 2, 3, 5, 7, 46, 19, 23  Intermediate 
116  1, 2, 3, 4, 5, 7, 10, 11, 13, 15, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 30, 34, 35, 36, 37, 39, 43, 45, 46, 51, 53, 56, 57, 58, 64  Intermediate 
117  2, 4, 37, 6, 7, 9, 10, 61, 48, 49, 55, 60, 42, 62, 5  Intermediate 
118  0, 1, 2, 3, 5, 7  Intermediate 
119  0, 1, 3, 40, 52, 54, 59, 60, 61, 62, 63  Intermediate 
120  0, 32, 34, 43, 38, 33, 11, 12, 14, 15, 16, 17, 21, 25, 24, 57, 26, 27, 28, 30, 31  Intermediate 
121  34, 4, 37, 38, 7, 63, 10, 43, 44, 48, 17, 20, 21, 54, 55, 56, 36, 26, 59, 58, 5  Intermediate 
122  0, 1, 3, 4, 8, 10, 13, 18, 19, 22, 23, 29, 35, 39, 40, 41, 45, 46, 47, 50, 52, 54, 59, 63  Intermediate 
123  0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 48, 49, 55, 42  Intermediate 
124  0, 1, 3, 8, 44, 46, 50, 19, 20, 23  Intermediate 
125  48, 59, 4, 6, 41, 10, 55, 47, 16, 49, 52, 9, 26, 15, 42  Intermediate 
126  0, 1, 2, 3, 4, 5, 7, 10  Intermediate 
127  1, 2, 3, 5, 6, 7, 9, 13, 18, 19, 22, 23, 29, 35, 36, 37, 39, 42, 45, 46, 48, 49, 51, 53, 55, 56, 58, 64  Intermediate 
128  0, 32, 38, 33, 12, 14, 21, 25, 28, 31  Intermediate 
129  1, 3, 6, 9, 13, 15, 16, 18, 19, 22, 23, 26, 29, 35, 36, 37, 39, 41, 42, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 58, 59, 64  Intermediate 
130  28, 18, 35, 12, 60  Intermediate 
131  2, 43, 5, 6, 7, 9, 11, 34, 15, 16, 17, 24, 57, 26, 27, 30  Intermediate 
133  0, 1, 2, 3, 4, 5, 7, 10  Intermediate 
134  0, 6, 8, 9, 12, 13, 14, 18, 19, 21, 22, 23, 25, 28, 29, 31, 32, 33, 35, 38, 39, 40, 41, 45, 46, 47, 50, 52, 54, 59, 63  Intermediate 
135  32, 2, 35, 5, 7, 60, 39, 12, 45, 14, 47, 49, 18, 22, 9, 28, 62  Intermediate 
136  3, 5, 6, 9, 10, 15, 16, 26, 37, 41, 43, 44, 45, 46, 47, 50, 52, 57, 58, 59, 63, 64  Intermediate 
137  4, 5, 7, 9, 10, 14, 17, 19, 20, 21, 22, 23, 26, 32, 34, 36, 38, 39, 43, 44, 45, 46, 47, 48, 49, 54, 55, 56, 58, 59, 62, 63  Intermediate 
138  1, 3, 11, 13, 15, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 30, 34, 35, 36, 37, 39, 43, 45, 46, 51, 53, 56, 57, 58, 64  Intermediate 
140  0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 15, 27, 30, 31, 33, 37, 42, 48, 49, 51, 53, 55, 57, 61, 64  Intermediate 
141  0, 1, 2, 3, 5, 6, 7, 9, 60, 61, 62  Intermediate 
142  0, 32, 38, 33, 12, 14, 21, 25, 28, 31  Intermediate 
143  0, 1, 2, 6, 8, 11, 12, 13, 15, 16, 18, 19, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 37, 40, 41, 42, 51, 52, 53  Intermediate 
144  0, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 38, 39, 41, 43, 45, 46, 47, 50, 57  Intermediate 
145  2, 43, 5, 7, 11, 34, 46, 15, 16, 17, 19, 23, 24, 57, 26, 27, 30  Intermediate 
146  0, 1, 3, 4, 6, 60, 10, 55, 12, 48, 49, 18, 35, 9, 28, 42  Intermediate 
147  0, 32, 59, 37, 38, 33, 31, 41, 12, 46, 47, 16, 19, 52, 14, 23, 25, 26, 15, 28, 21  Intermediate 
148  0, 1, 3  Intermediate 
149  37  Intermediate 
150  1, 3, 11, 13, 15, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 30, 34, 35, 36, 37, 39, 43, 45, 46, 51, 53, 56, 57, 58, 64  Intermediate 
151  2, 5, 7, 8, 44, 50, 20  Intermediate 
152  0, 1, 34, 3, 11, 15, 16, 17, 43, 24, 57, 26, 27, 30  N.D. 
153  0, 1, 3, 4, 5, 8, 10, 11, 13, 14, 17, 20, 21, 22, 24, 25, 29, 32, 34, 36, 38, 39, 40, 43, 44, 45, 46, 50, 54, 56, 57, 58, 63, 64  N.D. 
154  0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 60, 61, 62  N.D. 
155  0, 6, 9, 11, 12, 14, 15, 16, 17, 21, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 38, 42, 43, 48, 49, 55, 57  N.D. 
157  0, 1, 2, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 17, 20, 21, 22, 24, 25, 26, 29, 32, 34, 36, 37, 38, 39, 40, 41, 42, 47, 48, 49, 52,  
54, 55, 56, 59  Poor  
158  9, 10, 4, 6  Poor 
159  0, 1, 4, 8, 11, 13, 14, 17, 20, 21, 22, 24, 25, 29, 32, 34, 36, 38, 39, 40, 54, 56, 60, 61, 62  Poor 
160  0, 1, 4, 8, 11, 13, 14, 17, 20, 21, 22, 24, 25, 29, 32, 34, 36, 38, 39, 40, 54, 56  Poor 
161  10, 4  Poor 
162  0, 1, 3, 4, 37, 10  Poor 
163  32, 39, 9, 45, 14, 47, 49, 22, 62  Poor 
164  9, 11, 12, 14, 15, 16, 17, 18, 22, 24, 26, 27, 28, 30, 32, 34, 35, 37, 39, 40, 43, 45, 47, 49, 52, 54, 57, 59, 60, 62, 63  Poor 
165  37  Poor 
166  0, 1, 4, 6, 8, 9, 11, 13, 14, 17, 20, 21, 22, 24, 25, 29, 32, 34, 36, 38, 39, 40, 54, 56  Poor 
167  19, 46, 23  Poor 
168  59, 37, 6, 41, 47, 16, 52, 9, 26, 15  Poor 
169  9, 6  Poor 
170  0, 1, 4, 8, 11, 13, 14, 15, 17, 20, 21, 22, 24, 25, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 51, 53, 54, 56, 57, 61, 64  Poor 
171  60, 37, 62, 61  Poor 
172  10, 4  Poor 
174  46, 19, 23, 60, 61, 62  Poor 
175  0, 1, 4, 8, 11, 13, 14, 17, 19, 20, 21, 22, 23, 24, 25, 29, 32, 34, 36, 38, 39, 40, 46, 54, 56  Poor 
176  0, 1, 4, 6, 8, 9, 11, 13, 14, 15, 17, 20, 21, 22, 24, 25, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 51, 53, 54, 56, 57, 61, 64  Poor 
177  2, 5, 7, 8, 44, 46, 50, 19, 20, 23  Poor 
178  4, 6, 40, 9, 10, 61, 48, 49, 52, 54, 55, 59, 60, 42, 62, 63  Poor 
179  2, 5, 7, 9, 11, 14, 15, 16, 17, 22, 24, 26, 27, 30, 32, 34, 39, 43, 45, 47, 49, 57, 62  Poor 
180  2, 59, 4, 5, 7, 8, 41, 10, 44, 15, 16, 50, 52, 20, 26, 47  Poor 
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