Graph Theory - Graph Compression

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Graph Compression

Graph compression is the process of reducing the size of a graph while keeping its important structure and properties.

As graphs become larger and more complex, especially in real-world applications like social networks, transportation networks, and biological networks, it becomes important to find ways to store, process, and analyze them.

Graph compression helps by reducing memory usage, speeding up computations, and enabling transmission of large graphs over networks.

Why is Graph Compression Important?

Graph compression is important for several reasons −

• Memory Efficiency: Large graphs take up a lot of memory. Compression reduces this memory use, making it easier to store and work with big graphs.
• Faster Computations: Compressed graphs make it faster to process and analyze data, as they simplify the graph while keeping important details.
• Efficient Transmission: Compressed graphs are easier to send over networks, especially in systems like cloud computing or large distributed networks.
• Handling Big Data: Real-world graphs, like social networks and web graphs, can be huge. Compression helps us manage and process these large graphs efficiently.

Major Concepts in Graph Compression

To understand graph compression, it is important to know the following concepts −

• Graph Size: The size of a graph is measured by the number of nodes (points) and edges (connections). A larger graph uses more memory and takes more time to process.
• Graph Isomorphism: Two graphs are isomorphic if you can match their nodes and edges one-to-one without changing the way they are connected. Graph compression often involves finding similar subgraphs that can be stored more efficiently.
• Subgraphs: A subgraph is a part of the original graph made up of some nodes and edges. Finding repeating or similar subgraphs helps in compressing the graph.
• Compression Rate: The compression rate shows how much smaller the graph has become after compression, calculated as the ratio of the original size to the compressed size.

Types of Graph Compression

Graph compression can be categorized based on the compression techniques used −

• Lossless Compression
• Lossy Compression

Lossless Compression

Lossless compression means that the original graph can be perfectly restored from its compressed version. All the important properties of the graph are kept, and no information is lost during compression.

Techniques in lossless compression −

• Adjacency List Compression: This method stores the graph's connections (adjacency list) more efficiently using techniques like variable-length encoding, run-length encoding, or bitmaps.
• Graph Minors: This technique looks for subgraphs that repeat and stores them in a more compact form.
• Canonical Forms: The graph is represented in a unique, compressed form, so equivalent graphs are stored the same way.
• Delta Encoding: Instead of storing all edges, this method saves the differences between adjacent values, which works well when the graph has many similar edges.

Lossy Compression

Lossy compression reduces some information to achieve higher compression, which makes the graph much smaller. While it can shrink the graph's size, it may cause some details or properties to be lost.

This type of compression is useful when the exact original graph isn't needed and an approximation is okay.

Techniques in lossy compression −

• Edge Pruning: Removing edges that don't significantly affect the overall structure of the graph, like edges with very low weights in a weighted graph.
• Node Aggregation: Grouping similar or related nodes together into a single node, simplifying the graph.
• Graph Sampling: Picking a smaller, representative part of the graph (nodes and edges) that still captures the overall structure. This is helpful for large graphs that are too expensive to analyze fully.

Graph Compression Algorithms

There are several algorithms designed for graph compression, and each one focuses on different parts of the graph structure. Some of the most common algorithms are −

• Graphzip
• Graph Neural Network Based Compression
• MINCE (Minimal Compression Encoding)

Graphzip

Graphzip is a lossless algorithm that compresses a graph's adjacency list. It uses methods like run-length encoding, Huffman coding, and delta encoding to make the graph smaller while keeping all the information intact.

Following are the steps to compress a graph using the Graphzip algorithm −

• First, find repeated parts or patterns in the adjacency list.
• Use techniques like run-length or delta encoding to compress these repeated patterns.
• Store the compressed graph with the necessary information so it can be easily decompressed later.

Graph Neural Network Based Compression

Graph neural networks (GNNs) can be used to compress graphs by learning compact representations (embeddings) of nodes or subgraphs.

This method helps make large graphs smaller while still keeping important information. It is useful for tasks like classifying graphs, predicting nodes, or predicting connections between nodes.

Following are the steps to compress a graph using the GNN algorithm −

• Train a graph neural network to learn compact representations (embeddings) of nodes or the whole graph.
• Use these embeddings to represent the graph in a smaller, compressed form, while keeping the important structural details intact.

MINCE (Minimal Compression Encoding)

MINCE is a lossy graph compression method that reduces the size of the graph by combining similar nodes or edges.

It looks for less important edges, based on factors like edge weights or how connected nodes are, and removes them. This makes the graph smaller while keeping its main structure intact.

Applications of Graph Compression

Graph compression is used in various fields where large graphs are common, such as −

• Social Network Analysis
• Web Graph Compression
• Biological Networks
• Computational Biology and Drug Discovery

Social Network Analysis

In social networks, users are represented by nodes and relationships (like friendships or interactions) by edges. Graph compression helps store and process huge social networks efficiently while keeping important features like community structure and influence intact.

Web Graph Compression

The internet can be seen as a graph where web pages are nodes and hyperlinks are edges. Compression techniques help web crawlers store large web graphs more efficiently, speeding up search and indexing tasks.

Biological Networks

In biological networks, nodes can represent genes, proteins, or other biological elements, and edges represent interactions. Compression helps reduce memory and processing costs when analyzing large biological data, like protein interactions or gene networks.

Computational Biology and Drug Discovery

In computational biology, graphs represent molecular structures or biological pathways. Efficient compression helps store, analyze, and simulate complex biological systems, aiding in drug discovery and disease modeling.

Challenges in Graph Compression

Even though graph compression is helpful, it comes with several challenges −

• Scalability: Large graphs (like those in social networks or the web) require compression methods that can handle millions of nodes and edges efficiently.
• Loss of Information: With lossy compression, important details might be lost, which could affect the results of graph analysis.
• Dynamic Graphs: Real-world graphs are always changing, so compression methods must be flexible enough to add or remove nodes and edges easily.
• Compression Speed: Compressing and decompressing large graphs can take a lot of time, especially when using complex algorithms.