Data For Science: Graphs

Graph Neural Networks 101

Bruno Gonçalves — Sun, 12 Apr 2026 14:45:21 GMT

Welcome to the latest post in the Graphs for Data Science substack. In this post, we explore the fundamentals of Graph Neural Networks using a fascinating dataset on molecular structures.

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As always, you can find the companion notebook on the Graphs for Data Science GitHub repository:

Graphs For Data Science GitHub

In this post, we’re flipping our usual script, and instead of focusing on analyzing a graph, we’re going to learn from one. We’ll build a neural network to predict a molecule's solubility in water purely from the graph structure of the connections between atoms. The family of models that pulls this off goes by the name Graph Neural Networks (GNNs), and they’ve quietly become one of the most versatile tools in the modern ML toolkit.

For the sake of convenience (and a shiny new dataset), we’re using molecules as our playground. Still, the core ideas: message passing, neighborhood aggregation, and graph-level readout are applicable anywhere you encounter graph structure: social networks, road systems, knowledge…

k-Core Decomposition

Bruno Gonçalves — Sat, 24 Feb 2024 17:57:14 GMT

Welcome to the latest post in the Graphs for Data Science substack. In this post, we explore the structure of networks using k-core decomposition, a unique way to probe the connectivity patterns of complex networks.

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Real-world networks are typically large and hard to visualize, leading to the development of sophisticated methods and algorithms, such as degree correlations, network structure, and modularity, to explore their structure systematically. In this post, we continue this exploration by learning about k-core decomposition.

The k-core of a graph is the largest subgraph where every node has degree of at least k. The k-core can easily computed by recursively removing every node of degree less than k. In Python, this can done with:

Network Attacks

Bruno Gonçalves — Sun, 16 Oct 2022 23:27:47 GMT

Welcome to the latest post of Graphs for Data Science. This week we consider how to attack a network, even when we don’t have complete information about all the nodes and edges.

Thanks to node correlations, we can (preferentially) reach network hubs by selecting a node a random and then reaching out to one of their neighbors. We demonstrate the effectiveness of this approach by using a large real world social network and measuring the number of components as a function of the number of removed nodes.

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All of our posts so far, have focused on analyzing networks, their properties and function. Today, we take a different approach and explore how we can disrupt or break up a network without even needing to observe it completely.

As we saw in previous posts, real world networks display strong correlations among its nodes with edges being established preferentially towards some nodes in detriment of others. One indirect consequence of this is the fact that we can typically reach a hub if we si…

Kahn's Algorithm for Topological Sorting

Bruno Gonçalves — Wed, 21 Sep 2022 17:28:00 GMT

This week we explore Kahn’s algorithm for topological sorting of Directed Acyclical Graphs, a simple but powerful algorithm introduced in 1962 for the automatic scheduling of computational tasks while taking into account all of their dependencies. As a test case, we apply it to a large citation network dataset

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Topological Sorting

Topological Sorting is a sorting algorithm that produces a linear ordering of the nodes in a Direct Acyclic Graph that respects all the directions of the edges. In fact, one definition of a DAG is that of a graph that can be topologically sorted. A direct application of topological sorting is for scheduling computational tasks based on their dependencies. In this case, each node represents a task and a directed edges points from node i to node j if task i must be completed before task j.

Kahn’s Algorithm

One of the most famous topological sorting algorithms was introduced by A. B. Kahn in 1962. The fundamental idea is straightforward:

Identify all nodes with in-…

Prim's Minimum Spanning Tree Algorithm

Bruno Gonçalves — Tue, 16 Aug 2022 00:16:00 GMT

In this post we introduce Prim’s algorithm to identify the Minimum Spanning Tree of a graph. The MST is a tree made up by the subset of edges from a weighted graph that connect every node while having the smallest total weight possible.

MSTs have a wide range of practical applications, specially in distribution networks where we must reach every node in the graph while minimizing the overall cost. As a test case, we apply Prim’s algorithm to an Open Street Map road network.

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Prim’s Algorithm

Prim’s algorithm, despite its name, was originally introduced by V. Jarník in 1930 and eventually rediscovered independently by Robert Prim and Edsger W. Dijkstra in the late 50s.

The algorithm has some similarities with Dijkstra’s algorithm (that we discussed here) with a greedy procedure to iteratively grow the tree by adding one edge at a time.

Choose an arbitrary seed node.
At each step add the lightest edge that connects one of the nodes already in the tree to one of the nodes not yet added.
Stop w…

Neighborhood Overlap and Edge Weights

Bruno Gonçalves — Mon, 25 Jul 2022 03:14:00 GMT

In this post, we introduce an intuitive measure of topological overlap between two nodes. This metric allows us to have a clearer picture of the structure of the network, identify redundant nodes, “structural holes” and even the “strength of weak ties”.

We start by motivating the overlap with a simple toy network example before moving on to a real world social network.

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Network Overlap

In traditional physical systems we are often able to identify symmetries, parts of the system that are mirror images of other parts. For example, in a system like this:

Bipartite Graphs 101

Bruno Gonçalves — Tue, 28 Jun 2022 04:44:00 GMT

Today we have a quick introduction to Bipartite Networks. We introduce some of the fundamental concepts of one-mode network projection and explore how they can be used to better understand an actor-movie network based on Kaggle dataset.

We hope you enjoy today’s post. We’re looking forward to your comments, feedback and suggestions for follow ups. And of course, don’t forget to:

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Bipartite Networks

All the cases we’ve looked at so far have had just a single kind of node. While perhaps the most common, this is far from being the only possible case. Often we are interested in studying the relationship between two different types of nodes (say X-nodes and Y-nodes) where edges are allowed only between nodes of different kinds:

Epidemic Models

Bruno Gonçalves — Wed, 13 Apr 2022 21:22:00 GMT

Welcome to the latest post of the Graphs For Science substack, a cross post from the “Epidemic Modeling” series where I aim to guide you through the mathematical and conceptual details underlying epidemic modeling with a focus on what’s most relevant to the current CoVID-19 pandemic.

In this post, we explore some simple network models in order to better understand the role that degree correlations play in disease spreading. I aim to make each post self contained, but you might want to check out our previous posts to brush up on the basics of epidemic modeling:

Community Structure and Network Modularity

Bruno Gonçalves — Tue, 01 Mar 2022 04:11:00 GMT

In this post we build on several of our previous posts to introduce the concepts of Community Structure and Network Modularity. We implement in detail the Girvan-Newman algorithm and apply it to the NIPS co-authorship network.

We hope you enjoy today’s post. We’re looking forward to your comments, feedback and suggestions for follow ups. And of course, don’t forget to:

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Network Communities

Empirical networks can display a wide range of structures due, in part due to the underlying process that originated them. In particular, as we already saw in previous posts, networks can have properties multiple components, a non-trivial Degree Distribution, a high Clustering Coefficient, low Network Diameter, a high frequency of specific Network Motifs, etc.

All of these properties contribute to make the network look different from point to point. For example, specific groups of nodes in a network can be significantly denser than others with different dense regions being interconnected by sparsely popu…

Markov Chains and PageRank

Bruno Gonçalves — Sun, 05 Dec 2021 22:21:12 GMT

In this post we introduce the fundamental ideas underlying Markov Chains and how PageRank leverages them to obtain a measure of the relative importance of each node that is capable of providing an important new view on the structure of our network and the role that each node plays.

We hope you enjoy today’s post. We’re looking forward to your comments, feedback and suggestions for follow ups. And of course, don’t forget to:

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Markov Chains

We introduced the concept of a Random Walk in a previous post when we discussed Language Generation. The idea is a simple one: image you live on the nodes of a Graph and that at each time step you must follow a randomly chosen outgoing edges of your current node.

In this simple case, your next step depends only on your current position on the graph and not on other factors like how many times you’ve visited this how, how long since your last visits, etc. This is perhaps the simplest example of a class of Memoryless processes, where the next step depends …

Network Assortativity and the Configurational Model

Bruno Gonçalves — Sun, 17 Oct 2021 14:38:23 GMT

Our exploration of simple network models in previous posts has made it clear that nodes in real world networks are not connected at random but rather have well defined preferences in who they connect to.

In this weeks post, we explore the concept of degree-degree correlations in a network. Using simple metrics like the average degree of the nearest neighbors we explore how the structure and function of a network is reflected in the way it is wired.

To finalize, we introduce the Configurational Model that allows us to generate the complete ensemble of networks with the same degree distribution.

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Degree-Degree Correlations

We saw in our previous treatment of the Erdős-Rényi, Watts-Strogatz and Barabási-Albert models that connections between nodes are not created at random, resulting in a non-trivial degree distribution.

In this post we’ll explore a direct consequence of the rules governing the creation of edges in real word networks. We’ll use the BTS Airline Network dataset as our test cas…

Preferential Attachment and the Barabási-Albert model

Bruno Gonçalves — Sun, 05 Sep 2021 20:23:46 GMT

This week we finalize our exploration of some of the most fundamental theoretical models of networks with a look at the Barabasi Albert model.

The BA model helped to establish the concept of Preferential Attachment as the foundational mechanism underlying the power-law structure observed in empirical networks and helped to launch the field of Complex Networks that has proven to be a useful language and toolkit to study, model and understand empirical systems arising in fields as far flung as Linguistics, Epidemiology, Computer Science, Sociology, Economics and Physics.

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Broad-Tailed distributions

In previous posts, we’ve explored the Erdős-Rényi Model that generated random networks by connecting nodes at random, and the Watts-Strogatz Model that explained the small world phenomena by rewiring a regular network at random.

As successful as these two models were, they are still unable to explain one of the fundamental features of large-scale real-world networks, that of broad-tailed or scal…

The Watts-Strogatz Model and the Small World Effect

Bruno Gonçalves — Sat, 07 Aug 2021 17:44:35 GMT

This week we continue our exploration of basic network models with the Watts-Strogatz model and use it to introduce both the concept of Graph Diameter and the small world effect.

The Watts-Strogatz model was introduced in 1998 to directly address the limitations of the ER model and generate model networks that are closer to what we observe in the real world.

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Six-Degrees of Separation

One of the oldest websites on the internet is the Internet Movie Database, better known as IMDb, that was launched in 1990. The IMDb is the canonical reference for all things movie related such as reviews, cast members, revenue, etc. that currently includes 7.5 million titles (including episodes) and 10.4 million personalities in its database. It also has the distinction of being one of the first large (bipartite 1) network datasets made freely available online.

Erdős-Rényi Model and Clustering Coefficient

Bruno Gonçalves — Sun, 25 Jul 2021 03:41:08 GMT

While the main focus of this blog is the exploration and understanding of real world network datasets, it is also important to arm ourselves with a solid theoretical understanding of some of the fundamentals of Graph Theory and Graph Generation Algorithms.

This week we start our exploration of basic network models with the venerable Erdős-Rényi model for Random Graphs and use it to introduce the concept of the clustering coefficient both of which will come in handy in future posts.

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Erdős-Rényi Model

Perhaps the simplest of the network models, known as the Erdős–Rényi model of random graphs, was proposed in 1959 by two Hungarian mathematicians: Paul Erdős and Alfréd Rényi. Due to its simplicity and mathematical elegance, it spurred a surge of interest in Graph Theory by other mathematicians.

The fundamental ideal is very simple: to generate an ER network with N nodes you start with N nodes of degree zero and then attempt to add every possible edge connecting a pair of nodes with some fin…

Network Motifs

Bruno Gonçalves — Sun, 30 May 2021 19:10:25 GMT

The processes that generate the real world networks we are likely to analyze, produce characteristic signatures in the topology of networks. Common patterns in graphs are known as Network Motifs and can provide important information about the structure of our network and the process that generated it.

In this post we introduce the ESU algorithm to enumerate all the subgraphs of a network with a specific number of nodes and how we can use it to calculate the frequency of each network motif.

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Network Motifs

The networks we encounter in the real world are produced as the result of some natural or artificial process. The details of these processes can result in specific signatures within the structure of the graphs they produce.

To make this point clear, let us consider a simple example of just 3 nodes. There’s only two possible ways to connected 3 nodes among themselves in a way that results in a connected graph, namely:

Bitcoin Transaction Network

Bruno Gonçalves — Sun, 23 May 2021 03:31:14 GMT

This week we explore the Bitcoin transaction network, one of the fundamental data structures underlying the Bitcoin Blockchain

Bitcoin and other Blockchain based technologies have been in the news over the past couple of weeks due to wild swings in their prices and calls for stricter regulations but little attention has been given to what makes these technologies so interesting and widely applicable.

In this post we dive into the way in which the Bitcoin blockchain (the oldest and the one with the largest overall value) encodes and processes transactions and how we can apply some simple graph concepts to identify clusters of addresses that are controlled by the same user.

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We won’t be able to cover all the details of Bitcoin, but if you would like to truly understand the implementation of Bitcoin, I would highly recommend that you read Mastering Bitcoin and Programming Bitcoin.

BitCoin, introduced in 2008 by Satoshi Nakamoto in the aftermath of the 2008 financial crisis. Nakamoto…

Graph Components

Bruno Gonçalves — Sun, 25 Apr 2021 13:43:23 GMT

Real world networks come in all shapes and sizes, depending on the specific dataset we are dealing with. In particular, many networks of interest are disconnected, meaning that they can be separated into distinct subgraphs that have no edges connecting them.

In this post we explore a real world Twitter network and use it to better understand the various ways to define connected components.

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Connected Components

A graph is said to be connected if there is an Undirected Path connecting every pair of nodes. If some nodes are unreachable, then the network is said to be disconnected or have multiple components. Let us consider a simple example:

Node Centrality

Bruno Gonçalves — Sat, 03 Apr 2021 23:27:32 GMT

To understand the structure of a network, it is often useful to identify which nodes play the most important roles or are best placed within the network, and to quantify how central they are. Centrality measures and other graph metrics are often used as features used to train Machine Learning models.

In this post we explore some of the fundamental centrality measures we may use to characterize the relative importance of a node in the network: Degree Centrality, Closeness Centrality and Betweenness Centrality.

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In other to better illustrate the properties of each type of centrality, we start by creating a simple barbell-like network, with two Erdős–Rényi networks connected by a single node:

Graph Embeddings 101

Bruno Gonçalves — Sun, 28 Mar 2021 03:25:18 GMT

In this post we continue our exploration of Random Walks and Graph Traversal Algorithms and how they can be applied to help us understand our networks better.

We start by analyzing word2vec, a classic algorithm that is able to map words to numerical vectors that encode information about the meaning and similarity of words. This will lay the foundation for our first look at graph embedding algorithms with node2vec that leverages the basic structure of word2vec and some intuitions about random walks to generate meaningful vector representations of each node in an arbitrary graph.

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Word2vec

Introduced by Tomáš Mikolov, a Google Engineer in 2013, word2vec is an algorithm that tries to answer a deceptively simple question: can the meaning of a word be represented by a vector of numbers in such a way that words with similar meanings have similar vectors, or, at least, are located in the same general are of this vector space.

To achieve this, word2vec draws inspiration from the distributional hy…

Structural Causal Models

Bruno Gonçalves — Sun, 14 Mar 2021 02:25:05 GMT

Any systems or process that can be represented by interconnected components can be thought of as a Graph. For example:

Social Network — The Nodes are people and the Edges are friendship relationships
Internet — The Nodes are individual servers and the Edges are the cables connecting them
Airline Network — The Nodes are airports and the Edges are flight connections
Road Network — The Nodes are intersections and the Edges are streets
Ecosystem — The Nodes are individual species and the Edges the predator/prey relationships between them.

And many more examples could easily be found.

Not surprisingly, many Graphs are naturally directed, with edges specifically pointing from a source node to a target node:

Directed Graph — Each edge points only in one specific direction.

A particularly useful subset of directed graphs are those without any cycles, Directed Acyclic Graphs.

DAGs are extremely common when we are dealing with time ordered processes, in particular:

Trees - An important special case of D…