Modern Graph Theory Algorithms with Python: Harness the power of graph algorithms and real-world network applications using Python

Modern Graph Theory Algorithms with Python

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Many thanks and much gratitude to Peter Schnable, for his encouragement and guidance during my journey into science and mathematics; for long discussions about herpetology, conservation, and ethics; and for inspiring me to share my knowledge with others and to forge my own path in STEM for social good.

- Colleen Molloy Farrelly

To my beloved mother, Ngoy Justine, whose unwavering love and encouragement fueled my passion for knowledge. Throughout the writing of these pages, her memory has been a guiding light, inspiring me to pursue excellence and share the joy of discovery. Though she is no longer with us, her spirit lives on in the words and sentiments expressed within these chapters. With profound gratitude and love, I dedicate this work to the woman whose influence continues to shape my journey.

To Meda, Divine, and Abigael.

- Franck Kalala Mutombo

Foreword

As the CTO/CIO of life sciences as well as automotive, energy, and high-tech industry-focused companies, I have continuously been challenged with creating meaningful data insights from a great variety of unstructured and semi-structured data sources. A decade ago, my advanced analytics and machine learning journey experienced the biggest push forward when I benefited from one of the most impactful introductions in my career. Ever since I got to work with Colleen Farrelly on our ontology-focused data science requirements when building out a genomics diagnostics platform, my approach to AI has been tremendously elevated. Subsequently, I got to work on Colleen Farrelly’s exceptional book, The Shape of Data, which intrigued me deeply with her unique delivery of how to combine geometry and machine learning-based algorithms and supporting data structures to create powerful topological representations of complex data problems.

In this equally captivating sequel, Modern Graph Theory Algorithms with Python, Colleen Farrelly and Franck Kalala Mutombo take you to the next level of unleashing the potential of network science by diving deep into specific graph theory approaches to solve a great variety of industry problems ranging from ecological to financial, spatial and temporal sales, and clinical data challenges. It is easy to see why readers and data science practitioners of any level will find each chapter profoundly valuable based on the supporting Python library and code examples along with the underlying mathematical explanations.

With each turn of a page, I found myself wanting to reinforce the presented learnings by applying well-defined examples to my own business requirements. Being a keen user of graph databases, I very much enjoyed the hands-on practical discussions on specific technologies such as in Chapter 12

to illustrate the relative ease of Python integrations and built-in capabilities of open source solutions that can be leveraged in today’s growing powerful set of available tools.

This comprehensive book provides just-enough and just-in-time fundamental concepts to enable data scientists and software engineers to greatly elevate their machine learning techniques with directly applicable well-structured scenarios. Each of the consistent problem-solution breakdowns includes key considerations for the required data wrangling, transformation, and modeling aspects.

Furthermore, this book provides a convincing lead into the relevance of new frontiers such as quantum network science algorithms, neural network architectures as graphs, hierarchical networks, and hypergraphs. With a concise and easy-to-follow thought process, this book provides you with the important context of how to reduce the large volumes of parameterization required for large language models and address the critical aspect of metadata management via hypergraph databases, for example.

The use of graph theory today is highly relevant to every industry and science domain. Whether the challenge is to provide predictive modeling or simulations or the optimization of business operations or clinical outcomes and many more requirements, this book is an indispensable guide to mastering the complexities of these critical real-world challenges. With all the key insights and GitHub repository examples at your fingertips, you will be transformed instantly into a subject matter expert. The Modern Graph Theory Algorithms with Python exploration is a must-read and thoroughly enjoyable book.

Michael Giske

Chairman of Inomo Technologies and Global CIO of B-ON

Contributors

About the authors

Colleen Molloy Farrelly is a chief mathematician, data scientist, and researcher who has expertise in applying math to the biological, medical, social, and physical sciences. She has also authored the book, The Shape of Data. She has mentored, coauthored papers, and worked with people across Latin America, Africa, Europe, and Asia.

She is based in Miami, Florida in the US and holds a master’s in biostatistics from the University of Miami. She is passionate about educational initiatives in the developing world and speaks at conferences such as Women in Data Science, IEEE conferences, PyData, and Applied Machine Learning Days.

I want to thank the people who have supported me over the years, especially early on, including John and Nancy Farrelly, Peter Schnable, the Warmus family, Mr. and Mrs. De Jong, the Mayor families, Justin and Christy Moeller, Luke Robinson, and many professors and colleagues throughout my career.

To all my students and those who will come after me who motivate me to teach and share my knowledge, you can use math and science to change the world for the better.

Franck Kalala Mutombo is a professor of mathematics at Lubumbashi University and former academic director of AIMS-Senegal. He previously worked in a research position at the University of Strathclyde and AIMS-South Africa in a joint appointment with the University of Cape Town. He holds a PhD in mathematical sciences from the University of Strathclyde, Glasgow, Scotland. His current research considers the impact of network structure on long-range interactions applied to epidemics, diffusion, object clustering, differential geometry of manifolds, finite element methods for PDEs, and data science. Currently, he teaches at the University of Lubumbashi and across the AIMS Network.

I express gratitude to my supportive network throughout my journey. I’m thankful for friends and professors who’ve contributed to my career. I am indebted to the countless students across Africa and those who will succeed them. Their enthusiasm and curiosity serve as constant reminders of the profound impact that mathematics and science can have on shaping a better world. It is with gratitude that I embrace the opportunity to teach and share knowledge, fostering a community of learners committed to leveraging the transformative potential of these disciplines for the greater good.

About the reviewer

Casey Moffatt, with a master’s in applied mathematics and a double bachelor’s in pure mathematics and philosophy, specializes in graph theory, optimization, and computer science. He is proficient in Python and various essential software for graph data science, machine learning, and algorithm development. He is eager to push boundaries in mathematics and computer science. He would like to thank Packt Publishing and contributors for enabling projects like this and to the countless individuals behind open source technologies.

Table of Contents

Preface

Part 1: Introduction to Graphs and Networks with Examples

1

What is a Network?

Technical requirements

Introduction to graph theory and networks

Formal definitions

Creating networks in Python

Random graphs

Examples of real-world social networks

Other type of networks

Advanced use cases of network science

Summary

References

2

Wrangling Data into Networks with NetworkX and igraph

Technical requirements

Introduction to different data sources

Social interaction data

Spatial data

Temporal data

Biological networks

Other types of data

Wrangling data into networks with igraph

Social network examples with NetworkX

Summary

References

Part 2: Spatial Data Applications

3

Demographic Data

Technical requirements

Introduction to demography

Demographic factors

Geographic factors

Homophily in networks

Francophone Africa music spread

AIMS Cameroon student network epidemic model

Summary

References

4

Transportation Data

Technical requirements

Introduction to transportation problems

Paths between stores

Fuel costs

Time to deliver goods

Navigational hazards

Shortest path applications

Traveling salesman problem

Max-flow min-cut algorithm

Summary

References

5

Ecological Data

Technical requirements

Introduction to ecological data

Exploring methods to track animal populations across geographies

Exploring methods to capture plant distributions and diseases

Spectral graph tools

Clustering ecological populations using spectral graph tools

Spectral clustering on text notes

Summary

References

Part 3: Temporal Data Applications

6

Stock Market Data

Technical requirements

Introduction to temporal data

Stock market applications

Introduction to centrality metrics

Application of centrality metrics across time slices

Extending network metrics for time series analytics

Summary

References

7

Goods Prices/Sales Data

Technical requirements

An introduction to spatiotemporal data

The Burkina Faso market dataset

Store sales data

Analyzing our spatiotemporal datasets

Summary

References

8

Dynamic Social Networks

Technical requirements

Social networks that change over time

Friendship networks

Triadic closure

A deeper dive into spreading on networks

Dynamic network introduction

SIR models, Part Two

Factors influencing spread

Example with evolving wildlife interaction datasets

Crocodile network

Heron network

Summary

References

Part 4: Advanced Applications

9

Machine Learning for Networks

Technical requirements

Introduction to friendship networks and friendship relational datasets

Friendship network introduction

Friendship demographic and school factor dataset

ML on networks

Clustering based on student factors

Clustering based on student factors and network metrics

Spectral clustering on the friendship network

DL on networks

GNN introduction

Example GNN classifying the Karate Network dataset

Summary

References

10

Pathway Mining

Technical requirements

Introduction to Bayesian networks and causal pathways

Bayes’ Theorem

Causal pathways

Bayesian networks

Educational pathway example

Outcomes in education

Course sequences

Antecedents to success

Analyzing course sequencing to find optimal student pathways to graduation

Introduction to a dataset

bnlearn analysis

Structural equation models

Summary

References

11

Mapping Language Families – an Ontological Approach

Technical requirements

What is an ontology?

Introduction to ontologies

Representing information as an ontology

Language families

Language drift and relationships

Nilo-Saharan languages

Mapping language families

Summary

References

12

Graph Databases

Introduction to graph databases

What is a graph database?

What can you represent in a graph database?

Querying and modifying data in Neo4j

Basic query example

More complicated query examples

Summary

References

13

Putting It All Together

Technical requirements

Introduction to the problem

Ebola spread in the Democratic Republic of Congo – 2018-2020 outbreak

Geography and logistics

Introduction to GEEs

Mathematics of GEEs

Our problem and GEE formulation

Data transformation

Python wrangling

GEE input

Data modeling

Running the GEE in Python

Summary

References

14

New Frontiers

Quantum network science algorithms

Graph coloring algorithms

Max flow/min cut

Neural network architectures as graphs

Deep learning layers and connections

Analyzing architectures

Hierarchical networks

Higher-order structures and network data

An example using gene families

Hypergraphs

Displaying information

Metadata

Summary

References

Index

Other Books You May Enjoy

Preface

Hello there! Network science combines the power of analytics with the deep theoretical tools of graph theory to solve difficult problems in data analytics. This empowers researchers and industry engineers/data scientists to analyze data at scale and reframe intractable analytics problems to produce powerful insights into problems and predictions about system behaviors, including biological, physical, and social systems of interest.

There are many important applications of network science today, including these:

Social network data

Spatial data

Time series data

Spatiotemporal data

More advanced data structures, such as ontologies or hypergraphs

This book gives a brief overview of social network applications and focuses on the cutting edge of network science applications to areas of data science, such as transportation logistics, conversation, public health, linguistics, and education. By the end of your journey, you’ll be able to frame your own data problem within the framework of network science to derive insights and tackle difficult problems in your field.

We will provide the necessary mathematical background as we dive into practical examples and code related to our work in academia and industry over the past decades, including work on predicting Ebola outbreaks, forecasting food price volatility, modeling genetic and linguistic relationships, and mining social networks for insights into social tie formation. As the world faces food shortages, public health crises, economic inequality, supply chain breakdowns, and environmental crises, network science will play an important role in big data analytics for social good.

Who this book is for

This book is for you if you are working with data. To get the most out of the book, you should have some familiarity with Python, particularly the pandas and numpy packages. In addition, some familiarity with data analytics is assumed, though the network science tools and problems we tackle are built from scratch for readers without a background in those problems or methods.

Network science has a rich history in many scientific disciplines, including epidemiology, biomedical engineering, sociology, genetics, environmental science, particle physics, computer science, and economics. Its foundations in graph theory influence research in many areas of pure and applied mathematics as well. Anyone in the fields of science, technology, engineering, and mathematics can benefit from network science’s toolset and approach to problem-solving.

What this book covers

Chapter 1

, What Is a Network?, introduces the theoretical concept of a network and provides several examples of networks in real-world applications, including work with random graphs. We’ll also get started with Python’s igraph and NetworkX packages.

Chapter 2

, Wrangling Data into Networks with NetworkX and igraph, builds on Chapter 1

by providing three examples of real-world data that can be formulated as network data and showing how to convert data into network form in Python. We’ll introduce problems involving spatial data, temporal data, and spatiotemporal data and explore how network science can solve these problems by converting the data into network form.

Chapter 3

, Demographic Data, explores two real-world projects using demography data from the developing world to understand network structures and capacity for information/infectious disease spread. We’ll consider the demographic characteristics and network properties of a friend group to see how both types of information can influence disease spread.

Chapter 4

, Transportation Data, provides a real-world example of a transportation network and introduces tools related to minimum paths and network flow. We’ll consider optimal routing and the shortest paths to destinations, including multistop pathways from one location to another.

Chapter 5

, Ecological Data, shows a real-world example of an ecological network and introduces spectral graph theory tools, including spectral clustering and graph Laplacians.

Chapter 6

, Stock Market Data, examines a real-world example of stock market data analysis with network tools, including edge-based centrality measures of volatility. We’ll mine data for tipping points, heralding either a period of market growth or market crash.

Chapter 7

, Goods Prices/Sales Data, provides two real-world examples of commerce data analysis over both space and time with tools previously covered in time series and spatial data applications. We’ll examine sales and pricing trends across time and space to better understand consumer behavior and the impacts of pricing changes across time and space.

Chapter 8

, Dynamic Social Networks, introduces a real-world example of social network datasets evolving over time and analyzes their vulnerability to spreading processes, such as epidemics and misinformation flow. We’ll consider factors influencing ecological social networks’ vulnerability to spread of disease.

Chapter 9

, Machine Learning for Networks, presents a comprehensive description on network-based machine learning and deep learning, including examples with supervised, unsupervised, and semi-supervised learning to understand disease risks within social networks.

Chapter 10

, Pathway Mining, introduces Bayesian networks and mining for causal pathways using an educational data example, where we’ll see how course sequencing and performance influence student outcomes.

Chapter 11

, Mapping Language Families – an Ontological Approach, covers ontologies and maps between ontologies using a linguistic data example from the Nilo-Sudanic language family and its lexicon variations.

Chapter 12

, Graph Databases, introduces graph databases with Neo4j, including data from prior chapters and how to query Neo4j with graph tools introduced in prior chapters and Neo4j’s query language. We’ll see how graph databases and network science tools create synergy in data science, as well as efficient data storage solutions.

Chapter 13

, Putting It All Together, ties together material from previous chapters into a final project, analyzing spatiotemporal network data and demographic data from Ituri and North Kivu provinces with generalized estimating equations to understand the evolution of the 2019 Ebola epidemic.

Chapter 14

, New Frontiers, introduces quantum graph algorithms, graph theory for neural network optimization, hierarchical networks, and hypergraphs.

To get the most out of this book

We provide Python scripts and assume some knowledge of basic Python analytics packages (such as NumPy and scikit-learn) and Python syntax. We assume some knowledge of basic analytics tasks such as summary statistics and working with different types of data in Python with either numpy or pandas. Scripts are written for each chapter, with later scripts often depending on earlier scripts in the chapter to build knowledge. Other concepts in Python and in analytics will be introduced conceptually and then with Python code examples.

If you are using the digital version of this book, we advise you to type the code yourself or access the code from the book’s GitHub repository (a link is available in the next section). Doing so will help you avoid any potential errors related to the copying and pasting of code.

You are encouraged to try out the code examples in this book on your real-world data science projects. If you want to delve deeper into graph algorithms and network science, we encourage you to look at the latest research papers on network science topics. Google Scholar and arXiv are two good references for network science methods and application papers.

Download the example code files

You can download the example code files for this book from GitHub at https://github.com/PacktPublishing/Modern-Graph-Theory-Algorithms-with-Python

. If there’s an update to the code, it will be updated in the GitHub repository.

We also have other code bundles from our rich catalog of books and videos available at https://github.com/PacktPublishing/

. Check them out!

Conventions used

There are a number of text conventions used throughout this book.

Code in text: Indicates code words in text, database table names, folder names, filenames, file extensions, pathnames, dummy URLs, user input, and Twitter handles. Here is an example: This script shows that average subgraph centrality varies between the two subfamily trees, with Greenberg’s average subgraph centrality of 2.478 and Dimmendaal’s average subgraph centrality of 3.276.

A block of code is set as follows:

#compare subgraph centrality of language families

gs=nx.subgraph_centrality(G)

print(np.mean(np.array(list(gs.values()))))

gs2=nx.subgraph_centrality(G2)

print(np.mean(np.array(list(gs2.values()))))

Bold: Indicates a new term, an important word, or words that you see onscreen. For instance, words in menus or dialog boxes appear in bold. Here is an example: When you hover over the Movie DBMS label on the right-hand side of the screen, you’ll see a Start button that launches the connection to this database. Click on Start.

Tips or important notes

Appear like this.

Get in touch

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Part 1:Introduction to Graphs and Networks with Examples

This part of the book builds