Graph theory studies **networks of nodes and edges**—any structure where entities connect to one another. Originating in Euler’s 18th-century solution to the Königsberg bridges problem, the field now underpins computer science, epidemiology, social network analysis, linguistics, and neuroscience. At its core, graph theory asks: - **Which nodes are central?** (degree, betweenness, eigenvector centrality) - **How does information or influence flow?** - **Where do bottlenecks, clusters, or weak links occur?** - **How resilient is the network to failure or attack?** - **What patterns emerge from local interactions?** (small-world networks, scale-free networks) Graphs can be directed or undirected, weighted or unweighted, static or dynamic. The analytical power lies in revealing **structure** and **function** simultaneously: a network’s shape constrains what it can do. --- # **Connection to Integrated Information Theory (IIT)** Integrated Information Theory posits that consciousness corresponds to a system’s capacity to generate **integrated information** (Φ): information that is both highly differentiated and deeply unified. This has implicit network logic: 1. **Nodes as elements** IIT models systems as sets of elements with causal relationships—formally very close to **directed graphs**. 2. **Edges as causal links** Cause–effect relationships map naturally onto **weighted, directed edges**, which determine how information propagates. 3. **Integration and graph topology** High Φ requires a network that is neither a loose collection of modules nor a fully homogeneous mesh, but something in between: - richly **interconnected**, - yet **differentiated**, - resistant to partition. This is analogous to **small-world** or **complex** network topologies known to balance local clustering with global reach. 4. **Cuts and partitions** IIT’s mathematical procedure—identifying the “minimum information partition”—resembles evaluating **graph cuts** to measure how badly the system’s informational structure breaks when divided. 5. **Complex motifs** The theory’s emphasis on feedback loops, recurrent connections, and causal webs overlaps with graph theory’s study of **cycles**, **strongly connected components**, and **network motifs**. --- ### **In brief** Graph theory provides the **formal vocabulary**—nodes, edges, connectivity, partitions—while IIT gives a **philosophical and informational interpretation** of those structures. Both treat systems as networks whose **organisation** determines their **capabilities**, whether that capability is consciousness, communication, or resilience. `Concepts:` `Knowledge Base:`