Md. Asif Uddin

    Chapter 9 VII.9

    Biological Networks

    Protein, regulatory, metabolic and disease networks are four different graphs, and a centrality claim means something different in each.

    Four kinds of biological network, and what a topology claim can and cannot support.

    How this chapter is built

    M2Substantive

    The derivations are the chapter. A reader who skips the algebra has not learned it.

    basics3/9what the words mean
    concept2/2what to picture
    theory0/2why it works, and when it does not
    mathematics0/9derive it, then compute it
    practice0/6build it, break it, read the papers

    Five strands, not one. Mathematics is the spine; the other four are the body. A chapter cannot pay its way out of teaching with problems, nor out of problems with teaching.

    Before you start

    The problem

    Biology produces graphs at every scale. Their edges mean different things — binding, regulation, catalysis, comorbidity — and a statistic computed without that distinction is a number about a drawing rather than about a cell.

    Masking a convolution, and the collapse it causesOn the left, a masked grid with a kernel window straddling the boundary: an ordinary convolution reads across the hole. In the middle, sparse convolution treats visible patches as sparse data. On the right, channel responses collapse toward each other until global response normalisation forces them to compete.the problemFCMAEGRNa kernel spans the holeand leaks the answervisible patches only,as sparse datacollapsedcompetingchannels forced apartby normalisationNeither half works alone. Bolting MAE onto plain ConvNeXt made things worse.
    Fig. 9 — A kernel slides across a masked hole and leaks the answer; sparse convolution stops it, and global response normalisation stops the channels collapsing. Neither half works alone.

    What this chapter covers

    • Graphs
    • Protein interaction networks
    • Gene regulatory networks
    • Graph neural networks

    Apparatus

    The mathematics this chapter leans on, held in Book 0 so it can be assumed here without being taught here. Not a gate — follow a link when a step stops making sense.

    Graphs, vertices and edges 0.GR.01 · Representing a graph 0.GR.02 · Centrality and community structure 0.GR.09 · The identity, the inverse and the transpose 0.LA.02

    Notation

    • WA weight matrix
    • XA batch of token representations, T×d, rows are tokens
    • λA regularisation coefficient, or an eigenvalue where the context is linear algebra
    • dModel width

    Propositions

    Not yet written. The topics above are the plan for this chapter; each will become a proposition with its own figure.

    Worked problems

    0/3 problems0/3 variants0/6 exercisesowes 9 more

    Not yet written. At M2 this chapter owes 3 worked problems across 3 distinct variants, and 6 exercises, every one with a published solution. The build enforces that from the day the chapter is marked published.