mathematics//graph theory//nodal capacity

Why the same flow changes the state of two nodes at different speeds. The **nodal capacity** converts a transfer rate into a rate of change of the intensive variable. In a tank with volume \(V_i(h_i)\),

Why the same flow changes the state of two nodes at different speeds. The nodal capacity converts a transfer rate into a rate of change of the intensive variable. In a tank with volume Vi(hi)V_i(h_i)Vi​(hi​),

ai(hi)=dVidhi,V˙i=ai(hi)h˙i.a_i(h_i)=\frac{dV_i}{dh_i},\qquad \dot V_i=a_i(h_i)\dot h_i.ai​(hi​)=dhi​dVi​​,V˙i​=ai​(hi​)h˙i​.

For tanks of constant section, aia_iai​ is the cross-sectional area. Define M=diag⁡(ai)M=\operatorname{diag}(a_i)M=diag(ai​), the storage matrix. With linear conductances and inflows bbb,

Mh˙=−Lh+b,h˙=−M−1Lh+M−1b.M\dot h=-Lh+b,\qquad \dot h=-M^{-1}Lh+M^{-1}b.Mh˙=−Lh+b,h˙=−M−1Lh+M−1b.

LLL describes the connections (graph Laplacian); MMM the storage. In hydraulics, [h]=m[h]=\mathrm m[h]=m, [ai]=m2[a_i]=\mathrm m^2[ai​]=m2, [bi]=m3/s[b_i]=\mathrm m^3/\mathrm s[bi​]=m3/s and [kij]=m2/s[k_{ij}]=\mathrm m^2/\mathrm s[kij​]=m2/s.

Level and volume are not interchangeable. If v=Mhv=Mhv=Mh with constant MMM, then v˙=−LM−1v+b\dot v=-LM^{-1}v+bv˙=−LM−1v+b: the order of the matrices changes. It is not correct in general to use v˙=−M−1Lv\dot v=-M^{-1}Lvv˙=−M−1Lv, nor to call the state of h˙=−Lh\dot h=-Lhh˙=−Lh a volume without explaining capacities and units. In a closed network 1TMh\mathbf1^{\mathsf T}Mh1TMh, the total volume, is conserved, not necessarily ∑ihi\sum_i h_i∑i​hi​. The final level of a connected network is

h∗=∑iaihi(0)∑iai.h_*=\frac{\sum_i a_i h_i(0)}{\sum_i a_i}.h∗​=∑i​ai​∑i​ai​hi​(0)​.

The same scheme in other engineering. In a thermal network the state can be temperature and MMM the heat capacities; in an RC network, voltage and capacitances; the edges are thermal or electrical conductances. The resemblance comes from a balance and a law proportional to differences (balance equation), not from water, charge and heat being the same quantity.

Storage is not automatically mechanical inertia. Changing positive capacities does not by itself create oscillatory modes in this passive first-order network, because

M−1L=M−1/2(M−1/2LM−1/2)M1/2M^{-1}L=M^{-1/2}(M^{-1/2}LM^{-1/2})M^{1/2}M−1L=M−1/2(M−1/2LM−1/2)M1/2

and the central matrix is symmetric positive semidefinite, so the decay rates are real and non-negative. An individual level can rise and then fall by superposition of exponentials without any oscillatory mode. A mechanical model Mmq¨+Cmq˙+Lkq=0M_m\ddot q+C_m\dot q+L_kq=0Mm​q¨​+Cm​q˙​+Lk​q=0 is second order; without damping the modal problem Lkv=ω2MmvL_kv=\omega^2M_mvLk​v=ω2Mm​v yields frequencies, a different reading of eigenvalues from diffusion.

three-tank network compares capacities numerically; the Lyapunov function note proves dissipation for this network; the normalized Laplacian explains why dividing by degree is not the same as introducing an arbitrary physical capacity.