heatmap

Since most of these metrics already exist in networkx, you’ll just pull from there. You can check the networkx documentation for details.

community

features

for reference, gives an indication of how clustered the network is, and works by counting the proportion of triangles out of all edges that share a node.

triangles

statistics

103501035010^{350}

How do you define a distance between one node and another node?

largest

𝑆𝐼,𝐵𝐾),(𝐵𝐾,𝑀𝐻),(𝑀𝐻,𝑄),(𝑀𝐻,𝐵𝑋),(𝑄,𝐵𝑋

Island

𝐴

figure

see

degree

matrix

𝑑𝑒𝑔𝑟𝑒𝑒(𝑖)=∑𝑗=1𝑛𝑎𝑖𝑗=∑𝑗:𝑎𝑖𝑗=1𝑎𝑖𝑗+∑𝑗:𝑎𝑖𝑗=0𝑎𝑖𝑗=∑𝑗:𝑎𝑖𝑗=11+0

adjacency

𝑗jj

matrix

trite

In the context of this book, you will usually only worry about the undirected case, or when the presence of an arrow implies that the other direction exists, too. A network is undirected if an edge between node 𝑖ii and node 𝑗jj implies that node 𝑗jj is also connected to node 𝑖ii. For this reason, you will usually omit the arrows entirely, like you show below:

red

you look

The

Matrix

Laplacian

This lets you make inferences with 𝐿𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑LnormalizedL^{normalized} that you couldn’t make with 𝐿LL.

code

below

AttributeError Traceback (most recent call last)

4.1.2. The Incidence Matrix¶ Instead of having values in a symmetric matrix represent possible edges, like with the Adjacency Matrix, we could have rows represent nodes and columns represent edges. This is called the Incidence Matrix, and it’s useful to know about – although it won’t appear too much in this book. If there are 𝑛nn nodes and 𝑚mm edges, you make an 𝑛×𝑚n×mn \times m matrix. Then, to determine whether a node is a member of a given edge, you’d go to that node’s row and the edge’s column. If the entry is nonzero (111 if the network is unweighted), then the node is a member of that edge, and if there’s a 000, the node is not a member of that edge. You can see the incidence matrix for our network below. Notice that with incidence plots, edges are (generally arbitrarily) assigned indices as well as nodes. Click to show from networkx.linalg.graphmatrix import incidence_matrix cmap = sns.color_palette("Purples", n_colors=2) I = incidence_matrix(nx.Graph(A)).toarray().astype(int) fig, axs = plt.subplots(1, 2, figsize=(12,6)) plot = sns.heatmap(I, annot=True, linewidths=.1, cmap=cmap, cbar=False, xticklabels=True, yticklabels=True, ax=axs[0]); plot.set_xlabel("Edges") plot.set_ylabel("Nodes") plot.set_title("Incidence matrix", fontsize=18) ax2 = nx.draw_networkx(G, with_labels=True, node_color="tab:purple", pos=pos, font_size=10, font_color="whitesmoke", arrows=True, edge_color="black", width=1, ax=axs[1]) ax2 = plt.gca() ax2.text(.24, 0.2, s="Edge 1", color='black', fontsize=11, rotation=65) ax2.text(.45, 0.01, s="Edge 0", color='black', fontsize=11) ax2.set_title("Layout plot", fontsize=18) sns.despine(ax=ax2, left=True, bottom=True) Copy to clipboard /tmp/ipykernel_4749/905225500.py:4: FutureWarning: incidence_matrix will return a scipy.sparse array instead of a matrix in Networkx 3.0. I = incidence_matrix(nx.Graph(A)).toarray().astype(int) Copy to clipboard When networks are large, incidence matrices tend to be extremely sparse – meaning, their values are mostly 0’s. This is because each column must have exactly two nonzero values along its rows: one value for the first node its edge is connected to, and another for the second. Because of this, incidence matrices are usually represented in Python computationally as scipy’s sparse matrices rather than as numpy arrays, since this data type is much better-suited for matrices which contain mostly zeroes. You can also add orientation to incidence matrices, even in undirected networks, which we’ll discuss next. 4.1.3. The Oriented Incidence Matrix¶ The oriented incidence matrix is extremely similar to the normal incidence matrix, except that you assign a direction or orientation to each edge: you define one of its nodes as being the head node, and the other as being the tail. For undirected networks, you can assign directionality arbitrarily. Then, for the column in the incidence matrix corresponding to a given edge, the tail node has a value of −1−1-1, and the head node has a value of 000. Nodes who aren’t a member of a particular edge are still assigned values of 000. We’ll give the oriented incidence matrix the name 𝑁NN. Click to show from networkx.linalg.graphmatrix import incidence_matrix cmap = sns.color_palette("Purples", n_colors=3) N = incidence_matrix(nx.Graph(A), oriented=True).toarray().astype(int) fig, axs = plt.subplots(1, 2, figsize=(12,6)) plot = sns.heatmap(N, annot=True, linewidths=.1, cmap=cmap, cbar=False, xticklabels=True, yticklabels=True, ax=axs[0]); plot.set_xlabel("Edges") plot.set_ylabel("Nodes") plot.set_title("Oriented Incidence matrix $N$", fontsize=18) plot.annotate("Tail Node", (.05, .95), color='black', fontsize=11) plot.annotate("Head Node", (.05, 1.95), color='white', fontsize=11) ax2 = nx.draw_networkx(G, with_labels=True, node_color="tab:purple", pos=pos, font_size=10, font_color="whitesmoke", arrows=True, edge_color="black", width=1, ax=axs[1]) ax2 = plt.gca() ax2.text(.24, 0.2, s="Edge 1", color='black', fontsize=11, rotation=65) ax2.text(.45, 0.01, s="Edge 0", color='black', fontsize=11) ax2.set_title("Layout plot", fontsize=18) sns.despine(ax=ax2, left=True, bottom=True) ax2.text(-.1, -.05, s="Tail Node", color='black', fontsize=11) ax2.text(.9, -.05, s="Head Node", color='black', fontsize=11) Copy to clipboard /tmp/ipykernel_4749/2332982598.py:4: FutureWarning: incidence_matrix will return a scipy.sparse array instead of a matrix in Networkx 3.0. N = incidence_matrix(nx.Graph(A), oriented=True).toarray().astype(int) Copy to clipboard Text(0.9, -0.05, 'Head Node') Copy to clipboard Although we won’t use incidence matrices, oriented or otherwise, in this book too much, we introduced them because there’s a deep connection between incidence matrices, adjacency matrices, and a matrix representation that we haven’t introduced yet called the Laplacian. Before we can explore that connection, we’ll discuss one more representation: the degree matrix.

ax2.text(0, 0.2, s="Nodes 0 and 2 \nare connected", color='black', fontsize=11, rotation=63)

The most

1.1.2. Viewing Networks Statistically¶

1.1. What Is A Network?¶

features

For instance, each node might have a set of features attached to it: extra information that comes in the form of a table. For instance

Acknowledgements

models

The theoretical underpinnings of the techniques described in the book

network data science

Network Machine Learning and You¶

a statistical object

This book assumes you know next to nothing about how networks can be viewed as a statistical object

With the nodes as employees, edges as team co-assignment between employees, and node covariates as employee performance, you can isolate groups within your company which are over or underperforming employee expectations.

So

Representations of

, but are often more generalizable

Connection

𝑛nn nodes

this book

2

without

Feasible

𝜌ρ\rho

5.5.3. Correlated Network Models¶

𝑅(1)R(1)R^{(1)} and

score matrices

Let’s try this first by looking at the latent position matrices for 𝐀(1)A(1)\mathbf A^{(1)} and 𝐀(2)A(2)\mathbf A^{(2)} from the random networks for Monday and Tuesday first:

𝑁

𝑋⊤P=XX⊤P = XX^\top!

Pseudoinverse

array([0.74745577, 0.49019859])

f you think about it, however, you can think of this adjacency vector as kind of an estimate for edge probabilities. Say you sample a node’s adjacency vector from an RDPG, then you sample again, and again. Averaging all of your samples will get you closer and closer to the actual edge connection probabilities.

The average values were .775 and .149 - so, pretty close!

took a long time

𝐸𝑅ERER

their paper is called “On a two-truths phenomenon in spectral graph clustering” - it’s an interesting read for the curious)

- in 2018 -

Adjacency

useful information

them

Heck

Laplacian

two

Stochastic Block Model

Spectral Embedding

↓⎤⎦⎥⎥[←𝑢⃗ 𝑇1→]+𝜎2⎡⎣⎢⎢↑𝑢⃗ 2↓⎤⎦⎥⎥[←𝑢⃗ 𝑇2→]+...+𝜎𝑛⎡⎣⎢⎢↑𝑢⃗ 𝑛↓⎤⎦⎥⎥[←𝑢⃗ 𝑇𝑛→]

-

close

ninety-degree

Singular Value Decomposition

and another which rotates them bac

[x11"x22⋱"xnn]\begin{align*} \begin{bmatrix} x_{11} & & & " \\ & x_{22} & & \\ & & \ddots & \\ " & & & x_{nn} \end{bmatrix} \end{align*}

you’ll

submatrices

break a single matrix

default

Laplacians

The

started

The

5.2.7. Network models for networks which aren’t simple¶

whether you’re using a neural network, or a decision tree, or whether your goal is to classify observations or to predict a value using regression -

machine learning

6.3.1. The Coin Flip Example¶

𝑝∗p∗p^*

two plots look almost nothing alike

ℙ𝜃(𝐀=𝐴)=ℙ(𝐚11=𝑎11,𝐚12=𝑎12,...,𝐚𝑛𝑛=𝑎𝑛𝑛)=∏𝑖,𝑗ℙ𝜃(𝐚𝑖𝑗=𝑎𝑖𝑗)

(in network statistics, often just one)

we can never hope to understand the true distribution of 𝐀A\mathbf A.

⟨𝑥⃗ −𝑦⃗ ,𝑥⃗ −𝑦⃗ ⟩

With 𝑥⃗ x→\vec x defined as above in the sum example, we would have that: ∏𝑖=13𝑥𝑖=6.12∏i=13xi=6.12\begin{align*} \prod_{i = 1}^3 x_i = 6.12 \end{align*} if we were to use ={1,3}I={1,3}\mathcal I = \{1,3\}, then: ∏𝑖∈𝑥𝑖=3.4

ted to multiply all the elements of 𝑥⃗ x→\vec x, we would write: ∏𝑖=1𝑑𝑥𝑖=𝑥1×𝑥2×...×𝑥𝑑∏i=1dxi=x1×x2×...×xd\begin{align*} \prod_{i = 1}^d x_i = x_1 \times x_2 \times ... \times x_d \end{align*} Where ××\times is just multiplication like you are probably used to. Again, we have the exact same indexing conventions, where: ∏𝑖∈[𝑑]𝑥𝑖=∏𝑖=1𝑑𝑥𝑖=𝑥1×𝑥2×...×𝑥𝑑

In the general case, for a set S\mathcal S, we would say that 𝑆∈𝑟×𝑐S∈Sr×cS \in \mathcal S^{r \times c} if (think this through!) for any 𝑖∈[𝑟]i∈[r]i \in [r] and any 𝑗∈[𝑐]j∈[c]j \in [c], 𝑠𝑖𝑗∈sij∈Ss_{ij} \in \mathcal S

. We Like previously, there are two types of RDPGs: one in which 𝑋XX is treated as known, and another in which 𝑋XX is treated as unknown.

Symbolically

such

300300300

, since they are undirected; if we relax the requirement of undirectedness (and allow directed networks) 𝐵BB no longer need be symmetric.

simple

Imagine that we are flipping a fair single coin. A fair coin is a coin in which the probability of seeing either a heads or a tails on a coin flip is 1212\frac{1}{2}. Let’s imagine we flip the coin 202020 times, and we see 101010 heads and 101010 tails. What would happen if we were to flip 222 coins, which had a different probability of seeing heads or tails? Imagine that we flip each coin 10 times. The first 10 flips are with a fair coin, and we might see an outcome of five heads and five tails. On the other hand, the second ten flips are not with a fair coin, but with a coin that has a 4545\frac{4}{5} probability to land on heads, and a 1515\frac{1}{5} probability of landing on tails. In the second set of 101010 flips, we might see an outcome of nine heads and one tails. In the first set of 20 coin flips, all of the coin flips are performed with the same coin. Stated another way, we have a single group, or a set of coin flips which are similar. On the other hand, in the second set of twenty coin flips, twenty of the coin flips are performed with a fair coin, and 10 of the coin flips are performed with a different coin which is not fair. Here, we have two clusters of coin flips, those that occur with the first coin, and those that occur with the second coin. Since the first cluster of coin flips are with a fair coin, we expect that coin flips from the first cluster will not necessarily have an identical number of heads and tails, but at least a similar number of heads and tails. On the other hand, coin flips from the second cluster will tend to have more heads than tails.

happens

ER network

nework

Also, we let 𝐚𝑖𝑖=0aii=0\mathbf a_{ii} = 0, which means that all self-loops are always unconnected

When 𝑖>𝑗i>ji > j, we allow 𝐚𝑖𝑗=𝐚𝑗𝑖aij=aji\mathbf a_{ij} = \mathbf a_{ji}. This means that the connections across the diagonal of the adjacency matrix are all equal, which means that we have built-in the property of undirectedness into our networks

Erdös Rényi network

The Erdös Rényi model formalizes this relatively simple model with a single parameter:

model

If 𝐴AA and 𝐴″A″A'' are members of different equivalence classes; that is, 𝐴∈𝐸A∈EA \in E and 𝐴″∈𝐸′A″∈E′A'' \in E' where 𝐸,𝐸′E,E′E, E' are equivalence classes, then ℙ𝜃(𝐴)≠ℙ𝜃(𝐴″)Pθ(A)≠Pθ(A″)\mathbb P_\theta(A) \neq \mathbb P_\theta(A'').

(𝐴

𝐸EE

For the case of one observed network 𝐴AA, an estimate of 𝜃θ\theta (referred to as 𝜃̂ θ^\hat\theta) would simply be for 𝜃̂ θ^\hat\theta to have a 111 in the entry corresponding to our observed network, and a 000 everywhere else. Inferentially, this would imply that the network-valued random variable 𝐀A\mathbf A which governs realizations 𝐴AA is deterministic, even if this is not the case.

variaable

parameters

,

To

think

first

Fundamentally

Now, the only question is how to actually pull the separate latent positions for each network from this matrix.

big

rows

you don’t have to rotate or flip your points to line them up across embeddings.

big circle

bad

–

rotations of each other

none of the embeddings are rotations of each other

two

So

In this example they have eight (the number of columns in our combined matrix), but remember that in general we’d have 𝑚×𝑑m×dm \times d. T

averaging

great

node

he edges of

Remember

Grouping the Networks Separately¶

normal

Below, we turn our list of networks into a 3-D numpy array, and then do the above multiplication to get a new 3D numpy array of scores matrices. Because we embedded into two dimensions, each score matrix is 2×22×22 \times 2, and the four score matrices are “slices” along the 0th axis of the numpy array.

Any

Combining the networks¶

Combining the classifiers¶

Multiple Adjacency Spectral Embedding¶

communities

product

Model

Adjacency Spectral Embedding

For example, say you have nine networks of normal brains and nine networks of abnormal brains.

The normal and abnormal brains are all drawn from stochastic block models, but they come from different distributions. If you look at the code, you’ll see that the normal brains were set up to have strong connections within communities, whereas the abnormal brains were set up to have strong connections between communities. Below is a plot of the adjacency networks for every normal and every abnormal brain we’ve created.

types

create

algorithm

ℝ

(𝑥⃗ ⊤𝑖𝑥⃗ 𝑗

2

We will call the matrix 𝑃PP the probability matrix whose 𝑖𝑡ℎithi^{th} row and 𝑗𝑡ℎjthj^{th} column is the entry 𝑝𝑖𝑗pijp_{ij}, as defined above. Stated another way, 𝑃=(𝑝𝑖𝑗)P=(pij)P = (p_{ij})

𝑝

We showed above the model for the random node-assignment variable, 𝜏𝜏→ττ→\vec{\pmb \tau}, above.

process

𝑚11

R.V.

50

Likelihood

ℓ

𝜏⃗

known

Logically

unique

cluster

222

𝐸𝑖EiE_i,

1. and 2.

if it is totally unambiguous what 𝜃θ\theta refers to,

As there are no diagonal entries to our matrix 𝐴AA, this means that there are 𝑛2−𝑛=𝑛(𝑛−1)n2−n=n(n−1)n^2 - n = n(n - 1) values that are definitely not 000 in 𝐴AA, since there are 𝑛nn diagonal entries of 𝐴AA (due to the fact that it is hollow)

To summarize

𝐴 is an 𝑛×𝑛 matrix with 0s and 1s,𝐴 is symmetric,𝐴 is hollow

|||E||\mathcal E|

This means that all the networks are hollow.

right

learn

If you care about what’s under the hood you should also have a reasonable understanding of college-level math, such as calculus, linear algebra, probability, and statistics.

http://docs.neurodata.io/graph-stats-book/intro.html

Johns Hopkins University

for example

Network Machine Learning