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1 | 1 | # Mathematical Model
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2 | 2 |
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3 |
| -The core of the `NetworkDynamics.jl` package is the [`Network`](@ref) function. It accepts the functions describing the |
| 3 | +The core of the `NetworkDynamics.jl` package is the [`Network`]( |
| 4 | +@ref) function. It accepts the functions describing the |
4 | 5 | local dynamics on the edges and nodes of the graph `g` as inputs, and returns a composite function compatible with the
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5 | 6 | DifferentialEquations.jl syntax as output.
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6 | 7 |
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@@ -160,10 +161,12 @@ a flow *into* the connected vertex, whereas a negative flow represents a flow *o
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160 | 161 | └───────────────────┘
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161 | 162 | ```
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162 | 163 |
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163 |
| -### Single Sided Edge Outputs |
164 |
| -In cases where a conservation law is present in a system, the edge output functions $g_\mathrm{src}$ and |
165 |
| -$g_\mathrm{dst}$ are not independent, but rather one of them is a function of the other. For example, in a system with |
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| -a conservation law, the output at the source end is equal to the output at the destination end, i.e. $y_\mathrm{src} = -y_\mathrm{dst}$. |
| 164 | +### Single Sided Edge Outputs |
| 165 | +Systems exist, where the edge output functions $g_\mathrm{src}$ and $g_\mathrm{dst}$ are not independent, but rather one |
| 166 | +is a function of the other. For example, in a system with a flow conservation law, the flow magnitude at the |
| 167 | +source end is equal to the flow magnitude at the destination end (what flows in, must come out). Since the sign |
| 168 | +convention on both ends must be identical (e.g. positive flow is a flow towards the vertex) we get antisymmetric |
| 169 | +behavior: $y_\mathrm{src} = -y_\mathrm{dst}$. |
167 | 170 |
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168 | 171 | To accommodate such cases, we can use the concept of **single sided edge output functions**.
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169 | 172 | A single sided output function only defines a function for one of the outputs:
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