6.2 Stoichiometric matrix - Stoichiometry is the calculation of quantitative relationships of reactants and products in a balanced chemical reaction. Matrix: Reactions and flux across the top and metabolites down the side. Use methods of linear algebra (convex analysis) to analyze structural components of metabolic pathways.

6.3.1  & 6.3.2 Matrices – Adding, subtracting, and multiplying matrices is all material that is not new to me.

6.3.3 Linear Transformations, Ranks & Transpose – linear map is a functions between two vector spaces that preserves the operations of vector addition and scalar multiplication. Rank is a number that represents how linearly independent a matrix is. Use Gauss elimination method. Transpose: turns rows into columns and vice-versa.

6.3.4 Square Matrices & Matrix Inversion – square matrix, self explanatory. Inverse matrix times matrix = identity matrix.

Girven-Newman Algorithm

1. The betweenness of all existing edges is calculated.

2. The edge with the highest betweenness is removed.

3. The betweenness of all edges affected by the removal is calculated.

4. Repeat steps 2 and 3 until no edges remain.

4.7 Network Growth Mechanisms – models to represent how different networks grow and replicate. Duplication-Divergence Mechanism, duplication of vertex=duplication of gene. Duplication-Mutation-Complementation (DMC) based on model that most genes have been preserved by functional complementation. Duplication-Random Mutations (DMR). Some of this material really confused me, particularly how the edges were removed in the DMC algorithm. At least I understood how the edges changed with the addition of the duplicated vertices.

Hierarchical Clustering

For every pair i,j of vertices in the network, a weight Wij is computed that measures how closely connected the vertices are. Starting from the set of all vertices and no edges, edges are iteratively added between pairs of vertices in order of decreasing weight. Vertices are grouped into larger and larger communities, and the tree is built up to the root which represents the whole network.

Hierarchical clustering is often used together with other network measures to detect communities. Measures such as:

the number of vertex-independent graphs which are two paths that connect the same pair of vertices but have no common vertices other than the start and end points

the number of edge-independent graphs which share no common edges

total number of paths that run between i and j.

The number of paths between vertices i and j in a graph is equal to the minimum number of vertices/edges that must be removed from the graph to disconnect i and j and are therefore a measure of the robustness of the netowrk to deletion of vertices/edges. Problems with the hierarchical clustering method exist in single-edge-connected vertices where both the numbers of independent paths and the weighted path counts are small and thus the vertices often remain isolated from the network when communities are constructed.

Betweenness – Vertices that occur on many shortest paths between other vertices have a higher betweenness.

Edge betweenness is the number of shortest paths between pairs of vertices that run along a particular edge. Edges connecting communities will have high edge betweenness, and their removal would separate groups and reveal the underlying community structure of the graph.

Girven-Newman algorithm (2002)

1. Calculate betweenness for all m edges in a graph of n vertices (can be done in O(mn) time).

2. Remove the edge with the highest betweenness.

3. Recalculate betweenness for all edges affected by the removal.

4. Repeat from step 2 until no edges remain.

Because step 3 has to be done for all edges, the algorithm requires in the worst case O(m^2) operations.

4.3 Random Graphs – a graph of randomly placed edges. Average vertex degree is (n – 1)/p -> np. Random graphs are not adequate models for real-life networks due to a high tendency of clustering. These networks also tend to have right-skewed degree distributions that follow a power law, with a few massive outlying events.

4.4 Scale-Free Graphs – Degree distribution follows power law. Barabasi-Albert algorithm generates scale free networks. A node is more likely to receive additional links if already highly connected. Considered ultra small world. Party hubs interactwith most partners simultaneously, date hubs bind to partners at specific times.

4.1.1 Degree Distribution – Networks like the internet have fat tails with a low instance of exceptionally high outliers. Random graphs are roughly Poission. Directed graphs have an “In Degree” and “Out Degree.”

4.1.2 Clustering Coefficient – “The friend of my friend is also my friend.”  The ratio of the number of triangles / all possible triangles. A measure of how clustered a graph may be.

2.4.1 Description of Dijkstra’s Algorithm – used to find shortest path between two vertices in a directed graph with nonnegative edge weights.

Pseudocode – Meant to be read for human comprehension.

2.4.3 Running Time – “Cost” to run the algorithm. [constant x n = time]

2.2.1 Implementation of algorithms – (Algorithm) a finite set of precise instructions (steps) for performing a computation or for solving a problem. Speed and accuracy are most important. How does processing time increase as input increases?

1.5 Methods in Cellular Modeling – We will be modeling cellular activity using mathematical graphs, stoichiometric analysis, differential equations, and correlation functions.

~Crazy Ideas~

Sharks and networks?

Protein-protein interaction, “randomness” vs “structure” tension. Chaos – “small errors” diverge exponentially.