Introduction to Networks and Brain Networks

This lecture introduces the fundamentals of graph theory and network science, focusing on their application to brain networks. Starting with the historical breakthroughs in network science, such as Euler's Seven Bridges of Königsberg problem, it explores the importance of networks in complex systems, including the brain. The lecture covers the small-world and scale-free network discoveries, emphasizing the significance of connectivity patterns in various networks. It delves into the human connectome, inspired by the effort to map neural connections akin to the human genome project. The session concludes by highlighting the evolution of network neuroscience and the challenges faced in understanding the brain's intricate network structure.

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In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.

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Connectivity (graph theory)

In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v.

Path (graph theory)

In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges). A directed path (sometimes called dipath) in a directed graph is a finite or infinite sequence of edges which joins a sequence of distinct vertices, but with the added restriction that the edges be all directed in the same direction. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts.

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