One of the most popular statistics to use to determine sparsity in realized networks is the network density, but there are many others that have their own advantages [7], [8].

infinite

x→(n), this quantity could be written: 𝕩

ip:

element xi(n) is a random

9.3.1.3. The algorithmic implications#

Unfortunately, Fisher’s exact test has a slight caveat: it can be extremely computationally intensive to compute, especially when the number of data observations that we have (in this case, 200,000) is really big (it could be even bigger than 200,000).

SNP 1, alternative base T

Let’s assume that we have a small task, where for each row in the matrix X, we want to compute the row-wise sum. Stated another way, for a given row i, the quantity that you want to compute is ∑j=1mxij. If you ignore sparsity all-together, you can do this operation pretty easily: there are n rows, and m terms that you need to add together for each row, which means that you will have n⋅m total operations to perform (for each of n rows, perform an addition involving m terms).

If the rows can be sparse, the columns could be too; let’s assume that we have a matrix where m′ of the columns are sparse. Following a similar approach to the above, if we had a list Y with m′ elements telling us which columns were not sparse, we could just store the m′ non-sparse columns (each of which has n rows), and then the list of the m′ non-zero elements. Like above, we can store this information with 64⋅(n⋅m′+m′+1) bits.

Let’s say that of these n rows, we know ahead of time that a lot of the rows are sparse. By “row sparse”, what we mean is that xij=0 for all of these sparse rows i. Let’s assume that of the n total rows, only n′ are not sparse. We could, for instance, store the non-sparse rows in a little set X which has n′ elements telling us which rows are not sparse. For these non-sparse rows, we store all m pieces of column-wise information, but for the sparse rows, we just ignore them entirely. To store this entire matrix, we will need 64⋅(n′⋅m) (64 bits for each entry of a non-sparse row) +64⋅n′ (64 bits to store each element of X) +64 (to store the total number of rows that the matrix has), for a total of 64⋅(n′⋅m+n′+1) bits.

the rows are sparse. By “row sparse”, what we mean is that xij=0 for all of these sparse rows i.

but a common cutoff is if the number of non-zero elements is at most the number of rows or columns.

L’Hopital’s

Examples

intuitively

allows

attributes

mentally

Networks with cross-network attributes

For

with more than one element

usually

whether or not the approach can be used in isolation from a statistical model (non-model based or model-based network learning systems).

and

As the internet became widespread and coding tools became easier to use – Python became prevalent in machine learning, for instance, and cloud computing came into its own with Amazon’s AWS and Microsoft’s Azure –

cloud computing came into its own with Amazon’s AWS and Microsoft’s Azure –

One crucially influential application for networks was in 1996, when a graduate student at Stanford named Larry Page made the PageRank algorithm. The idea was that websites on the internet (which, in 1996, had barely formed) could be ordered into a hierarchy by “link popularity”: a web page would rank higher the more links there were to it. Larry Page and his friend Sergey Brin realized that PageRank could be used to create a search engine – and so they used the PageRank algorithm to found a small web searching company they called Google.

machine

Fig

who could potentially have the mental illness

network

1.2.2.3. We might errorfully observe the networks¶

, and although this book doesn’t focus on GNNs specifically, it does give you the fundamental ideas that you can build off of to understand them.

organized

Broadly

Dr

independence

Microsoft

ericwb95 - at - gmail - dot - com

Doksum

texts

would be

we think a reasonable

learning

which

unfortunately

Machine Learning

easy to use

everything unique

Twitter

nearly

the development of machine learning strategies for data that is a network.

For

have

We don’t really like that word

machine learning

nd each column represented the length and biological sex (male or female) of the lobster

a piece of

wikipedia

Learning

Hands-on Network Machine Learning with Scikit-Learn and Graspologic¶

SignalSubgraph

10.2.3.2. Classification with Bayes Plugin Classifier¶

10.2.3.1. Bayes Plugin Classifier (Statistical Intuition)¶

humans

astronauts

Can we come up with a signal subnetwork classifier?

astronauts

lobes

Estimation

above

There will be the same number of adjacency matrices as there are time points, since our network will be changing over time.

implement

VNSGM

8.4

-

Unshuffling

match ratio(𝑃,𝑃𝑢)

match_ratio

𝑃𝐵PBPB

reorder

0,1,2,3}

The

If we consider the worst possible case (every edge in 𝐴AA does not exist in 𝐵BB), 𝐴=012012⎛⎝⎜⎜011101110⎞⎠⎟⎟𝐵=012012⎛⎝⎜⎜000000000⎞⎠⎟⎟𝐴−𝐵=012012⎛⎝⎜⎜011101110⎞⎠⎟⎟||𝐴−𝐵||2𝐹=6

𝐴=012012⎛⎝⎜⎜011101110⎞⎠⎟⎟𝐵=012012⎛⎝⎜⎜011101110⎞⎠⎟⎟𝐴−𝐵=012012⎛⎝⎜⎜000000000⎞⎠⎟⎟||𝐴−𝐵||2𝐹=0

package

stochastic_block_test

familywise error rate

𝐵(

special

This

.8

multipletests

𝑎≠𝑏a≠ba \neq b

assumptions

Let’s formalize this situation a little bit more. We have the following three hypotheses. 𝐻0:𝑝1=𝑝2=𝑝3=𝑎H0:p1=p2=p3=aH_0: p_1 = p_2 = p_3 = a, against 𝐻1:𝑝1=𝑝2=𝑎H1:p1=p2=aH_1: p_1 = p_2 = a, but 𝑝3=𝑐p3=cp_3 = c. Finally, we have 𝐻2:𝑝1=𝑎H2:p1=aH_2: p_1 = a, 𝑝2=𝑏p2=bp_2 = b, and 𝑝3=𝑐p3=cp_3 = c. The hypothesis 𝐻HH is nested in the hypothesis 𝐻′H′H' if whenever 𝐻HH is true, 𝐻′H′H' is also true. In this sense, the hypothesis 𝐻′H′H' is said to contain the hypothesis 𝐻HH. Let’s consider 𝐻0H0H_0 and 𝐻1H1H_1, for instance. Notice that if 𝐻0H0H_0 is true, then 𝑝1=𝑝2=𝑝3=𝑎p1=p2=p3=ap_1 = p_2 = p_3 = a. However, 𝐻1H1H_1 is also true, since 𝑝1=𝑝2=𝑎p1=p2=ap_1 = p_2 = a, and 𝑝3=𝑐p3=cp_3= c can also be set equal to 𝑝1p1p_1 and 𝑝2p2p_2 if 𝑐=𝑎c=ac = a. A sequence of hypotheses 𝐻0,𝐻1,...,𝐻𝑛H0,H1,...,HnH_0, H_1, ..., H_n is called sequentially nested if 𝐻0H0H_0 is nested in 𝐻1H1H_1, which is nested in 𝐻2H2H_2, so on and so forth up to 𝐻𝑛−1Hn−1H_{n-1} is nested in 𝐻𝑛HnH_n. Note that the sequence of hypotheses that we presented for our three coin example are sequentially nested. We already saw that 𝐻0H0H_0 was nested in 𝐻1H1H_1. Now, let’s compare 𝐻2H2H_2 to 𝐻1H1H_1. Notet that if 𝑎=𝑏a=ba = b, that 𝑝1=𝑝2p1=p2p_1 = p_2, and 𝑝3=𝑐p3=cp_3 = c, exactly as in 𝐻1H1H_1, so 𝐻1H1H_1 is nested in 𝐻2H2H_2. Therefore, since 𝐻0H0H_0 is nested in 𝐻1H1H_1 and 𝐻1H1H_1 is nested in 𝐻2H2H_2, The sequence 𝐻0H0H_0, 𝐻1H1H_1, and 𝐻2H2H_2 are sequentially nested.

samples with which we are presented

and

presenting

describe

=

faithful

Pretty exciting, huh?

overcoming

. Unfortunately, if the data is not well-summarized by a normal distribution, the 𝑡tt-test tends to be a fairly poor choice for hypothesis testing.

8.2.2.2.2. Weighted Networks¶

,

below plot

8.2. Testing for Differences between Groups of Edges¶

the

the number of adjacencies in cluster one with an adjacency of zero

8.2.2.1. Hypothesis Testing with coin flips¶

alternative

indicates

RDPG

8.2.1. The Structured Independent Edge Model is parametrized by a Cluster-Assignment Matrix and a probability vector

higher chance two students are friends if they go to the same school than if they go to two different schools.

resort

the

8.1.1.2. Evaluating

8.1.1.1

heatmap

hat if your true labels are disproportionate

You

Temporary cluster assignments

enters from previous iteration

3 step

smack dab

ry to find reasonable guesses at the “centers”

dataset

ur goal is to learn about the block matrix, 𝐵BB,

these nodes tend to be more connected (more edges exist between and amongst them)

Non-Identifiability

had to delete

and so, f

Embedding

humans + aliens

and so forth

forth

first

used

However, as you can see, the colors are flipped: the communities are in different places relative to each other.

plot_latents

one

P = np.array([[pa, pb], [pc, pd]]) return sbm([n, n], P, return_labels=return_labels) # make nine human networks # and nine alien networks p1, p2, p3 = .12, .06, .03

because

bilateralized

you’ll

you’ll just simulate

simply

aving less stuff to deal with

Advisors

dependence

wheel

statsmodels

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Spectral Embedding

5.6.1

RDPG

Adm

IER

easily

coins

Ranking

networks

normalization

Sparsification

Lowering

done

Note

exclude the diagonal

bias

he task easier to estimate

The bias/variance tradeoff is

Ignoring

degree

4.4.1. Regularization of the Nodes

Degree

You

Nodes

space

Embedding this new matrix will give us a point in space for each network.

dissimilarity

All you need to get out of this code is that you have six networks from the first group, and another twelve networks from the second.

the whole

Nodes plotted \nas 2d points

on a coordinate axis

Euclidean

moving

Euclidean

issue

statsmodels

end

outlier

outlier

that

you’d

f you’re familiar with correlation, you’ll notice that these correlation numbers generally have a pretty high magnitude: each feature generally tells you a lot about each other feature.