diff --git a/content/algorithms/assortativity/correlation.md b/content/algorithms/assortativity/correlation.md
index c5aec79d..fa7a3867 100644
--- a/content/algorithms/assortativity/correlation.md
+++ b/content/algorithms/assortativity/correlation.md
@@ -5,7 +5,7 @@ jupytext:
extension: .md
format_name: myst
format_version: 0.13
- jupytext_version: 1.13.8
+ jupytext_version: 1.14.2
kernelspec:
display_name: Python 3
language: python
diff --git a/content/algorithms/dag/index.md b/content/algorithms/dag/index.md
index 6022e17f..f02281d2 100644
--- a/content/algorithms/dag/index.md
+++ b/content/algorithms/dag/index.md
@@ -5,7 +5,7 @@ jupytext:
extension: .md
format_name: myst
format_version: 0.13
- jupytext_version: 1.13.8
+ jupytext_version: 1.14.2
kernelspec:
display_name: Python 3
language: python
diff --git a/content/algorithms/euler/euler.md b/content/algorithms/euler/euler.md
index f8ff2ac9..eb1777e7 100644
--- a/content/algorithms/euler/euler.md
+++ b/content/algorithms/euler/euler.md
@@ -5,7 +5,7 @@ jupytext:
extension: .md
format_name: myst
format_version: 0.13
- jupytext_version: 1.13.6
+ jupytext_version: 1.14.2
kernelspec:
display_name: Python 3 (ipykernel)
language: python
diff --git a/content/algorithms/flow/dinitz_alg.md b/content/algorithms/flow/dinitz_alg.md
index 9df18d89..4ae67fc0 100644
--- a/content/algorithms/flow/dinitz_alg.md
+++ b/content/algorithms/flow/dinitz_alg.md
@@ -5,7 +5,7 @@ jupytext:
extension: .md
format_name: myst
format_version: 0.13
- jupytext_version: 1.13.8
+ jupytext_version: 1.14.2
kernelspec:
display_name: Python 3 (ipykernel)
language: python
diff --git a/content/algorithms/index.md b/content/algorithms/index.md
index cd233db9..c941e4a3 100644
--- a/content/algorithms/index.md
+++ b/content/algorithms/index.md
@@ -1,5 +1,10 @@
+
++++
+
++++
+
# Algorithms
A closer look at some of the algorithms and network analysis techniques
@@ -13,4 +18,5 @@ assortativity/correlation
dag/index
flow/dinitz_alg
euler/euler
+sub_isomorphism/subgraph_iso_img
```
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new file mode 100644
index 00000000..2268ab10
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new file mode 100644
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new file mode 100644
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--- /dev/null
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+---
+jupytext:
+ text_representation:
+ extension: .md
+ format_name: myst
+ format_version: 0.13
+ jupytext_version: 1.14.2
+kernelspec:
+ display_name: Python 3 (ipykernel)
+ language: python
+ name: python3
+---
+
+# Subgraph Isomorphism: Real world applications
+
++++
+
+First let's define the Subgraph Isomorphism problem:
+
+Let $H$ and $G$ be graphs. $H$ is a subgraph of $G = (V, E)$ if
+
+$\exists \hspace{0.1cm}G_o = (V_o, E_o)$ / $V_o \subseteq V \wedge E_o \subseteq E \cap V_o \times V_o \land G_o \hspace{0.1cm}$ and $H$ are isomorphic
+
++++
+
+Don't be scared of the math formula we can break this up to fully understand the problem.
+
+The main idea of this definition is that $H$ is a subgraph isomorphism of $G$ if we can find in $G$ a graph $G_o$ that is isomorphic with $H$. For $G_o$ to be in $G$ we need the nodes and edges in $G_o$ to be present in $G$. Formally, We need that $V_o$ (the nodes of $G_o$) to be a subset of $V$ (the nodes of $G$). Similarly, we need to ask that $E_o$ (the edges of $G_o$) are a subset of the edges of $G$, and the edges in $E_o$ only involve nodes that are present in $V_o$.
+
+This can be better understood with an example. Let's build a particular case of $G$, $G_o$ and, $H$.
+
+```{code-cell}
+import matplotlib.pyplot as plt
+import numpy as np
+import networkx as nx
+
+# Create a lollipop graph
+G = nx.lollipop_graph(5, 2)
+
+# Create a graph that is a subgraph of G
+G_o = nx.complete_graph(5)
+
+# Create a graph H that is isomorphic with G_o
+H = nx.complete_graph(["a", "b", "c", "d", "e"])
+
+plt.subplot(131)
+plt.title("G", fontweight="bold")
+# Define colors for the nodes of G: the nodes in sub_G are blue and the others are yellow
+colors = ["c" for i in range(0, 5)] + ["y" for i in range(5, 7)]
+nx.draw_spring(G, with_labels=True, node_color=colors) # Plot G
+
+plt.subplot(132)
+
+plt.title("Subgraph of G($G_o$)", fontweight="bold")
+nx.draw_circular(G_o, with_labels=True, node_color="c") # Plot sub_G
+
+plt.subplot(133)
+plt.title("H", fontweight="bold")
+nx.draw_circular(H, with_labels=True, node_color="m") # Plot H
+```
+
+Many real-world structures can be represented using graphs. Some of them are involved in problems that can be solved using sub-graph isomorphism algorithms. In this notebook we will study two real-world applications:
+- Pattern Recognition in images
+- Proteins classification by structure
+
++++
+
+## Where's my phone?
+
++++
+
+Let's use subgraph isomorphism to find our phone in our messy desk. First, these are the images of our desk and phone:
+
+```{code-cell}
+plt.subplot(121)
+m = plt.imread("img/nocopy_150x100.png")
+plt.imshow(m)
+plt.title("Messy desk")
+
+plt.subplot(122)
+c = plt.imread("img/phone_color.png")
+plt.imshow(c)
+plt.title("Indestructible phone")
+plt.show()
+```
+
+Image Source:
+
++++
+
+To simplify this problem, let's convert these images into grayscale. After this, the images will be a 2d array where each position represents a pixel.
+Also, let's define 0 (black pixel) as a null pixel. So we will have to delete all null pixels from our desk image. This will let us easily delete the background in the phone image.
+
+```{code-cell}
+def rgb2gray(rgb):
+ return np.dot(rgb[..., :3], [0.2989, 0.5870, 0.1140])
+```
+
+```{code-cell}
+# Preprocessing: Delete black pixels
+I = rgb2gray(plt.imread("img/nocopy_150x100.png"))
+for i in range(0, I.shape[0]):
+ for j in range(0, I.shape[1]):
+ if I[i][j] == 0:
+ I[i][j] = 0.0001
+plt.imsave("img/nocopy_150x100_p.png", I)
+```
+
+These will be the final images:
+
+```{code-cell}
+plt.subplot(121)
+
+I = rgb2gray(plt.imread("img/nocopy_150x100_p.png")) # Desk
+plt.imshow(I, cmap="gray")
+plt.title("Messy desk")
+
+plt.subplot(122)
+
+I2 = rgb2gray(plt.imread("img/phone.png")) # Object to find
+plt.imshow(I2, cmap="gray")
+plt.title("Indestructible phone")
+plt.show()
+```
+
+### Using Graphs to represent images
+
++++
+
+For this, we can use *grid_2d_graph(n, m)*. This function creates a $nxm$ grid where each coordinate is a node and each node is connected to its four nearest neighbors.
+
+For the phone graph, we will delete all the nodes corresponding to null pixels as they represent the background of the image.
+
+```{code-cell}
+desk_graph = nx.grid_2d_graph(I.shape[0], I.shape[1])
+phone_graph = nx.grid_2d_graph(I2.shape[0], I2.shape[1])
+
+# Define the node attribute weight as pixel values in desk_graph
+for i in range(0, I.shape[0]):
+ for j in range(0, I.shape[1]):
+ desk_graph.nodes[(i, j)]["weight"] = I[i][j]
+
+# Define the node attribute "weight" as pixel values in phone_graph
+# and delete useless nodes
+for i in range(0, I2.shape[0]):
+ for j in range(0, I2.shape[1]):
+ if I2[i][j] != 0:
+ phone_graph.nodes[(i, j)]["weight"] = I[i][j]
+ else:
+ phone_graph.remove_node((i, j))
+```
+
+Let's see how the phone_graph looks like coloring the nodes.
+
+```{code-cell}
+import matplotlib.cm as cm
+from matplotlib.colors import Normalize, rgb2hex
+
+# Match values to colors in RGB
+def color_map_color(value, cmap_name="gray", vmin=0, vmax=1):
+ norm = Normalize(vmin=vmin, vmax=vmax)
+ cmap = cm.get_cmap(cmap_name)
+ rgb = cmap(norm(abs(value)))[:3]
+ color = rgb2hex(rgb)
+ return color
+
+
+# Get RGB colors of the phone photo
+I2_colors = []
+for i in range(0, I2.shape[0]):
+ for j in range(0, I2.shape[1]):
+ if I2[i][j] != 0: # ignore null pixels
+ I2_colors.append(color_map_color(I2[i][j]))
+
+nx.draw_kamada_kawai(phone_graph, node_color=I2_colors)
+```
+
+Finally, let's see if we can match our phone graph in the desk graph. For this, we'll use the function *subgraph_is_isomorphic()* from the isomorphism module. To consider the node attribute "weight" in the matching we need to use the function *numerical_node_match(attribute, tol)*.
+
+```{code-cell}
+import networkx.algorithms.isomorphism as iso
+
+# Match node weight attributes with tolerance 10^-10
+em = iso.numerical_node_match("weight", 10 ^ -10)
+
+g = iso.GraphMatcher(desk_graph, phone_graph, node_match=em)
+
+g.subgraph_is_isomorphic()
+```
+
+We could successfully match the phone graph with the desk graph. This means that the phone graph is a subgraph isomorphism of the desk graph. More formally, this means that there's a possible matching function between the nodes of the phone graph and the nodes of the desk graph. We could use the matching to locate the nodes corresponding to the phone in the desk graph then the node labels are the pixel coordinates in the original picture.
+
++++
+
+Also, we can check that the desk and phone graphs are not isomorphic.
+
+```{code-cell}
+nx.is_isomorphic(desk_graph, phone_graph, node_match=em)
+```
+
+## Proteins Secondary structure
+
++++
+
+Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins are key in many biological processes within organisms.
+
+Proteins differ from one another primarily in their sequence of amino acids, which is dictated by the nucleotide sequence of their genes, and which usually results in protein folding into a specific 3D structure that determines its activity.
+
+Biochemists often refer to four distinct aspects of a protein's structure. We will be interested in the secondary and tertiary structure.
+
+**Secondary Structure** : regularly repeating local structures stabilized by hydrogen bonds. The most common examples are the α-helix, β-sheet, and turns. Because secondary structures are local, many regions of different secondary structures can be present in the same protein molecule.
+
+```{code-cell}
+plt.imshow(plt.imread("img/2_structure.png"))
+plt.title("Secondary Structure", fontweight="bold")
+plt.axis("off")
+plt.rcParams["figure.figsize"] = (2, 2)
+plt.show()
+```
+
+Image Source:
+
++++
+
+**Tertiary Structure** : is the 3D shape of a protein. This structure have information of the interactions between the R groups of the amino acids that make up the protein. Let's see how some proteins tertiary structure looks like:
+
+```{code-cell}
+plt.rcParams["figure.figsize"] = [7, 7]
+
+plt.subplot(121)
+plt.imshow(plt.imread("img/1fn3.png"))
+plt.title("1FN3 Tertiary Structure", fontweight="bold")
+plt.axis("off")
+
+plt.subplot(122)
+plt.imshow(plt.imread("img/1crn.png"))
+plt.title("1CRN Tertiary Structure", fontweight="bold")
+plt.axis("off")
+plt.show()
+```
+
+Image Source:
+
++++
+
+Many graph representations can be built from the Secondary and Tertiary Structure. Each representation capture different information about the protein. For example,
+- Use α-helix and β-sheet as nodes and the interaction energy between them to define edges.
+
+- $C_\alpha$ Networks: $C_\alpha$ atom of an amino acid residue is considered a node and an edge are drawn if the $C_\alpha$ distance between a pair of residues is within a threshold distance. $C_\alpha$ atoms are in the $\alpha$-helix structure. It is one of the simplest and most widely analyzed protein contact networks that captures very well the 3D topology of protein structure.
+
++++
+
+We will work on $C_\alpha$ networks. We will read the networks from edgelists.
+
+Data Source:
+
+Let's see graphs of proteins with PDB codes 1CRN(Plant protein), 1FN3(oxygen storage/transport in the human body) and 1EGJ(Plant Protein).
+
+```{code-cell}
+p_1CRN = nx.read_edgelist("data/1CRN_edgelist.txt", nodetype=str)
+nx.draw_spring(p_1CRN, node_color="g")
+plt.title("1CRN Plant Protein", fontweight="bold")
+plt.show()
+```
+
+```{code-cell}
+p_1FN3 = nx.read_edgelist("data/1FN3_edgelist.txt", nodetype=str)
+nx.draw_spring(p_1FN3, node_color="r")
+plt.title("1FN3 oxygen storage/transport Protein", fontweight="bold")
+plt.show()
+```
+
+```{code-cell}
+p_1EJG = nx.read_edgelist("data/1EJG_edgelist.txt", nodetype=str)
+nx.draw_spring(p_1EJG, node_color="g")
+plt.title("1EJG Plant Protein", fontweight="bold")
+plt.show()
+```
+
+We can identify graphlets (induced subgraphs) that are present in these graphs and use that to classify proteins. We can extract some subgraphs from the proteins and test if they are present in other proteins. Let's find a subgraph of 1CRN that is also a subgraph of 1EJG but not of 1FN3. This is interenting because 1CRN and 1EJG are both plant proteins but 1FN3 is not. But clearly to decide if this graphlet is particular of plant proteins we should test it in more proteins.
+
+```{code-cell}
+# Get a induced subgraph from 1CRN
+graphlet = p_1CRN.subgraph(["A" + str(i) for i in range(0, 30)])
+nx.draw(graphlet)
+
+# Test if the graphlet is a subgraph of each protein
+g1 = iso.GraphMatcher(p_1CRN, graphlet)
+print("Graphlet present in 1CRN ", g1.subgraph_is_isomorphic())
+
+g2 = iso.GraphMatcher(p_1FN3, graphlet)
+print("Graphlet present in 1FN3", g2.subgraph_is_isomorphic())
+
+g3 = iso.GraphMatcher(p_1EJG, graphlet)
+print("Graphlet present in 1EJG", g3.subgraph_is_isomorphic())
+```
+
+On a similar way we can find a graphlet in 1FN3 that is not present in 1CRN and 1EJG.
+
+```{code-cell}
+# Get a induced subgraph from 1FN3
+graphlet = p_1FN3.subgraph(["A" + str(i) for i in range(100, 110, 1)])
+plt.rcParams["figure.figsize"] = (5, 5)
+nx.draw(graphlet)
+
+# Test if the graphlet is a subgraph of each protein
+g1 = iso.GraphMatcher(p_1CRN, graphlet)
+print("Graphlet present in 1CRN ", g1.subgraph_is_isomorphic())
+
+g2 = iso.GraphMatcher(p_1FN3, graphlet)
+print("Graphlet present in 1FN3", g2.subgraph_is_isomorphic())
+
+g3 = iso.GraphMatcher(p_1EJG, graphlet)
+print("Graphlet present in 1EJG", g3.subgraph_is_isomorphic())
+```
+
+Another option is to use randomly generated graphs given a certain number of nodes, for example, using Erdos-graphs generators. *erdos_renyi_graph(n, p, seed)* generates a graph of n nodes in which all possible edges are added with probability p. We can find a random graphs that is present in 1FN3 but not in 1CRN and 1EJG.
+
+```{code-cell}
+random_graph = nx.erdos_renyi_graph(7, 0.9, seed=8)
+nx.draw(graphlet)
+
+# Test if the random graph is a subgraph of each protein
+g1 = iso.GraphMatcher(p_1CRN, random_graph)
+print("Random graph present in 1CRN ", g1.subgraph_is_isomorphic())
+
+g2 = iso.GraphMatcher(p_1FN3, random_graph)
+print("Random graph present in 1FN3", g2.subgraph_is_isomorphic())
+
+g3 = iso.GraphMatcher(p_1EJG, random_graph)
+print("Random graph present in 1EJG", g3.subgraph_is_isomorphic())
+```
+
+Also there are some graphlets and random graphs that are present in all proteins, for example:
+
+```{code-cell}
+random_graph = nx.erdos_renyi_graph(5, 0.9, seed=8)
+nx.draw(graphlet)
+
+# Test if the random graph is a subgraph of each protein
+g1 = iso.GraphMatcher(p_1CRN, random_graph)
+print("Random graph present in 1CRN ", g1.subgraph_is_isomorphic())
+
+g2 = iso.GraphMatcher(p_1FN3, random_graph)
+print("Random graph present in 1FN3", g2.subgraph_is_isomorphic())
+
+g3 = iso.GraphMatcher(p_1EJG, random_graph)
+print("Random graph present in 1EJG", g3.subgraph_is_isomorphic())
+```
+
+This technique can have many applications, for example, building tree-classification models. But in order to find graphlets and random graphs that are useful to clasify proteins it's important to test multiple proteins and also identify if those graphs are more present in some proteins than in random graphs.
+
++++
+
+### References
+-
+-
+-
+- Chakrabarty B. and Parekh N., "NAPS: Network Analysis of Protein Structures", Nucleic Acids Research, 2016, Vol. 44, Web Server issue W375–W382
+
+- Vishveshwara S.,Brinda K. V. and Kannan N., "Protein Structure: Insights from graph theory", Journal of Theoretical and Computational Chemistry, Vol. 1, No. 1 (2002) 000–000.
+- Henneges C., Röttig M., Kohlbacher O., Zell A., "Graphlet data mining of energetical interaction patterns in protein 3D structures", ICFC-ICNC 2010 - Proceedings of the International Conference on Fuzzy Computation and International Conference on Neural Computation, Valencia, Spain, October 24-26, 2010
+
diff --git a/content/exploratory_notebooks/facebook_notebook.md b/content/exploratory_notebooks/facebook_notebook.md
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extension: .md
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format_version: 0.13
- jupytext_version: 1.13.8
+ jupytext_version: 1.14.2
kernelspec:
display_name: Python 3
language: python
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extension: .md
format_name: myst
format_version: 0.13
- jupytext_version: 1.13.8
+ jupytext_version: 1.14.2
kernelspec:
display_name: Python 3
language: python
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+++ b/content/generators/index.md
@@ -1,5 +1,10 @@
+
++++
+
++++
+
# Graph Generators
A closer look at the functions provided by NetworkX to create interesting
diff --git a/content/generators/sudoku.md b/content/generators/sudoku.md
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extension: .md
format_name: myst
format_version: 0.13
- jupytext_version: 1.13.8
+ jupytext_version: 1.14.2
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display_name: Python 3 (ipykernel)
language: python