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consistently use SCC abbreviation
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cameronraysmith committed Jul 22, 2015
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Expand Up @@ -208,8 +208,7 @@ \subsection{Quantification of network hierarchy}
\subsection{Symmetries of SCCs and their relationship to robustness}
This map $\hier$ is many-to-one and so there is a large class of
operations which leaves $\hier(G)$ invariant for a given graph $G$ \reffighiertransformations{}. These three symmetries, \reffigscc{}, represent transformations that can be performed on the interaction graph that do not change the network of SCCs to which it is associated.
For instance, we may interchange the positions of the strongly
connected components relative to each other \reffighiertransformations$a$. Leaving the components fixed, we may move links between nodes in a component \reffighiertransformations$b$ or between components, or even add or delete links \reffighiertransformations$c$.
For instance, we may interchange the positions of the SCCs relative to each other \reffighiertransformations$a$. Leaving the components fixed, we may move links between nodes in a component \reffighiertransformations$b$ or between components, or even add or delete links \reffighiertransformations$c$.

Symmetries with respect to some property of the system are characterized by the ability to interchange these modules or their connectivity without changing that property. Two of these three intrinsic symmetries of $\hier$ are also symmetries with respect to dynamical robustness. \ref{fig:robustnesssymmetries} shows an example of these latter symmetries applied to a specific interaction graph. The collections of networks that have equivalent dynamical robustness characterized by its symmetries allow for a classification of neutral networks with respect to any process in which maximal robustness is selected for.

Expand Down Expand Up @@ -240,7 +239,7 @@ \subsection{Decomposition of stability over SCCs}

\subsection{General computation of dynamical robustness}

To relate the robustness of a graph to the robustness of its SCCs, we substitute \ref{eq:singleresamplemutau} and \ref{eq:stabfactor} into \ref{eq:robustnessbygraph}, collect factors corresponding to components, decompose integrals into their respective products, collapse integrals over delta distributions, and cancel common factors between numerator and denominator. Let $L$ denote the set of edges of $G$ that connect distinct strongly connected components. Then, if $i \in C_\alpha$ for some SCC $C_\alpha$, we have
To relate the robustness of a graph to the robustness of its SCCs, we substitute \ref{eq:singleresamplemutau} and \ref{eq:stabfactor} into \ref{eq:robustnessbygraph}, collect factors corresponding to components, decompose integrals into their respective products, collapse integrals over delta distributions, and cancel common factors between numerator and denominator. Let $L$ denote the set of edges of $G$ that connect distinct SCCs. Then, if $i \in C_\alpha$ for some SCC $C_\alpha$, we have
\begin{widetext}
\begin{equation}\label{eq:robustncessforsccs}
\begin{aligned}
Expand Down Expand Up @@ -343,7 +342,7 @@ \subsection{Simultaneous perturbation of multiple interaction strengths}
\end{widetext}
Then, given $(m_0, m_1, \ldots, m_n) \in M$, there are ${m \choose
m_0, m_1, \ldots. m_n}$ ways of choosing $m_i$ links from $C_i$ and
$m_0$ links between strongly connected components. Hence, our
$m_0$ links between SCCs. Hence, our
weighted average becomes
\begin{widetext}
\begin{equation}\label{eq:robustnessmultiple}
Expand All @@ -358,8 +357,7 @@ \subsection{Simultaneous perturbation of multiple interaction strengths}

As before, since $R(C_\alpha, S, \mu, \tau^{(m_i)}) \le 1$, we may
increase $R(G, S, \mu, \tau^{(m)})$ by increasing the maximum
possible value of $m_0$ while keeping the strongly connected
components the same. Again, if we fix $\hier(G)$, the maximum
possible value of $m_0$ while keeping the SCCs the same. Again, if we fix $\hier(G)$, the maximum
possible value of $m_0$ is $\sum_{(\alpha,\beta) \in \hier(G)} v_\alpha v_\beta$
whereas, if we allow it to vary, the maximum is $\frac{1}{2} ((\sum_{\alpha=1}^n
v_\alpha)^2 - \sum_{\alpha=1}^n v_\alpha^2)$, which is attained when $\hier(G_{max}) =
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