text:
\section{$\Ntheta$ and the Casson invariant}
\label{sec:Casson}

In this section, we prove Theorem~\ref{thm:Casson}.  At the heart of
this computation is a surgery formula.

We focus presently on the case where $Y$ is an integral homology
three-sphere. In this case, there is only one $\SpinC$ structure, and
we denote its invariant by $\Ntheta(Y)$. Note that $\Ntheta(Y)$ is
an integer, since in this case, the APS correction term $\Corr(Y)$ is
an integer for all metrics.

Let $Y$ as above and fix a knot $K\subset Y$. Let $Y_{p/q}$ be the
manifold obtained from $Y$ by $(p/q)$-surgery along $K$, so that $Y_0$
is an integral homology $S^1\times S^2$, and $Y_1$ is another integral
homology three-sphere. Note that, since $b_1(Y_0)=1$, the theory
from~\cite{Theta} applies, to give us an integer-valued invariant
$\theta(Y_0,\spinc)$ for each $\SpinC$ structure
$\spinc\in\SpinC(Y_0)$.

Our first result is the following surgery formula for $\Ntheta$.

\begin{theorem}
\label{thm:SurgeryFormula}
$$\Ntheta(Y_1)-\Ntheta(Y)=\sum_{\spinc\in\SpinC(Y_0)}\theta(Y_0,\spinc).$$
\end{theorem}

This theorem, together with Theorem~\ref{Theta:thm:CalculateZ}
from~\cite{Theta}, which relates $\theta(Y_0)$ with the Alexander
polynomial of $Y_0$, gives the following:

\begin{theorem}
Let $A=a_0+\sum_{i=1}^k a_i(T^i+T^{-i})$ be the symmetrized Alexander
polynomial of $Y_0$, normalized so that
$$A(1)=1.$$ Then,
$$\Ntheta(Y_1)-\Ntheta(Y)=\sum_{j=1}^{\infty} j^2 a_j.$$
\end{theorem}

\begin{proof}
We recall from~\cite{Theta} that $\theta(Y_0,\spinc)=\sum_{j=1}^\infty
j a_{|i|+j}$, where the $\SpinC$ structure $\spinc$ corresponds to the
integer $i$ under the identification $\SpinC(Y_0)\cong H^2(Y;\Z)\cong
\Z$ which sends the spin structure to $0$.
\end{proof}

\begin{remark}
It is an easy consequence of this theorem that:
$$\Ntheta(Y_{1/n})-\Ntheta(Y)=n\sum_{j=1}^{\infty} j^2 a_j.$$
\end{remark}

\begin{cor}
Let $Y$ be an oriented homology three-sphere. Then, $2\Ntheta(Y)$ is
equal to the Casson invariant $\lambda(Y)$.
\end{cor}

\begin{proof}
Every integral homology three-sphere can be obtained from $S^3$ by a
sequence of $\pm 1$ surgeries. So, it follows that $\NInv$ is uniquely
determined by its surgery formula and its value on $S^3$.  It is easy
to see that $\NInv(S^3)=0$: fix a genus one Heegaard decomposition of
$S^3$, and let $\met_t$ be a constant family of metrics on the torus,
which is clearly an allowable path, and indeed $\Inv(S^3)=0$ for this
path.  Furthermore, the correction terms cancel each other by
Proposition~\ref{prop:CompareCorrections}.

Since Casson's invariant $\lambda$ satisfies the same surgery formula
(see~\cite{Casson}), and $\lambda(S^3)=0$ as well, we get the result.
\end{proof}

To prove Theorem~\ref{thm:SurgeryFormula}, we find a Heegaard
decomposition $Y=U_0\cup_{\Sigma} U_1$ for which the knot $K$ is dual to
the last attaching disk for $U_1$, i.e. it intersects the last
attaching disk transversally in one point, and is disjoint from the
other ones.  To see that this can be arranged, start with a Morse
function on $Y-\nbd{K}$ with $1$ zero-handle, $g$ one-handles, and
$g-1$ two-handles. This gives us a handlebody $U_0$ and $g-1$
two-handles, whose attaching circles we denote by
$\{\beta_1,...,\beta_{g-1}\}$. Completing the Morse function over
$\nbd{K}$, we have described the second handlebody $U_1$ so that $K$
is dual to the final attaching disk. Heegaard decompositions for the
various surgeries $Y_{p/q}$ are given by surgeries in $U_1$: thus,
these can be described by fixing one surface $\Sigma$, one complete
set of attaching circles $\{\alpha_1,...,\alpha_g\}$, a $g-1$-tuple of
attaching circles $\{\beta_1,...,\beta_{g-1}\}$, and allowing the
final attaching circle $\beta_g$ to vary.

We would like to compare the $\theta$-invariants for the various
surgeries. To that end, we fix a path $\met_t$ of metrics, so that
$\met_0$ is $U_0$-allowable and $\met_1$ is $U_1$ allowable for any
choice of $\beta_g$. Such a metric $\met_1$ can be found, thanks to
Lemma~\ref{Theta:lemma:MissThetaHS} of~\cite{Theta}, which shows that
any metric which is sufficiently stretched out normal to the
$\{\beta_1,...,\beta_{g-1}\}$ is $U_1$-allowable.

We define a ``moduli space'' belonging to the knot complement and the
path of metrics.
$$\ModSp_{\met_t}(Y-K)=\{(D,s,t)\in\Sym^{g-1}\times[0,1]\times[0,1]\big|
s\leq t,
\Theta_{\met_s}(D)\in L(U_0),
\Theta_{\met_t}(D)\in\Lambda(U_1)\},$$
where $\Lambda(U_1)\subset \Jac$ is the set of connections $B\in\Jac$
so that $\Hol_{\beta_i}(B)=0$ for $i=1,...,g-1$.  There is a
map $$\rho\colon \ModSp_{\met_t}(Y-K)\longrightarrow
\Torus{2},$$
which is analogous to a boundary value map, given by measuring
holonomy around the meridian $m$ and the longitude $\ell$ of $K$,
i.e.  $$\rho(D,s,t)=\Hol_{m\times\ell}(\Theta_{\met_t}(D)),$$
normalized so that the point $0\times 0 \in \Torus{2}$ corresponds to
the spin structure which extends over $Y$.  With these conventions,
then, any ``reducible'' (in the sense of
Definition~\ref{def:Reducible})
on $Y-K$ can be restricted to
the boundary; its holonomies will lie in the circle $S^1\times\{0\}$.

Note that $\Lambda(U_1)\cap \Theta_{\met_1}(\Sym^{g-1}(\Sigma))$ is
not necessarily empty. However, for any $\epsilon>0$, there is a $T$
so that if $\met_1$ is stretched out at least $T$ normal to the
$\{\beta_1,...,\beta_{g-1}\}$, then the holonomy of any point in
$\Lambda(U_1)\cap \Theta_{\met_1}(\Sym^{g-1}(\Sigma))$ around
$m\times\ell$ lies in an $\epsilon$ neighborhood of
$\OneHalf\times\OneHalf$. This follows from
Lemma~\ref{Theta:lemma:MissThetaHS} of~\cite{Theta}.

This moduli space has the following important properties:

\begin{prop}
\label{prop:StructureModSpace}
For any $\epsilon>0$, if $\met_1$ is sufficiently stretched out normal
to the attaching circles $\{\beta_1,...,\beta_{g-1}\}$, then the
moduli space of the knot complement $\ModSp_{\met_t}(Y-K)$ is
generically a compact, smooth, one-dimensional manifold with two types
of boundary components corresponding to $t=1$ and $s=t$.  Furthermore,
the $t=1$ boundary maps under $\rho$ into an $\epsilon$-neighborhood
of $\OneHalf\times\OneHalf$, and those with $s=t$ map under $\rho$ to
$S^1\times 0$.
\end{prop}

\begin{proof}
Smoothness follows from the generic metrics statement
(Proposition~\ref{prop:GenericMetrics}). There is no $s=0$ boundary
since the metric $\met_0$ is $U_0$-allowable. The $t=1$ boundary lies in
$\Lambda(U_1)\cap \Theta_{\met_1}(\Sym^{g-1}(\Sigma))$, which maps near
$\OneHalf\times\OneHalf$, as above. The $s=t$ boundary corresponds to
the intersection of the theta divisor with $\Lag(U_0)\cap
\Lambda(U_1)$, which in turn maps to $S^1\times 0$. Indeed, this
latter circle corresponds to the circle of reducibles
$Y-K$:
the point $0\times 0$ corresponds to the spin structure
which extends over $Y$, while the point $\OneHalf\times 0$ corresponds
to the spin structure on $Y_1$.
\end{proof}

As we shall see presently, the $\theta$-invariants for the surgered
manifolds $Y$, $Y_1$, and $Y_0$ are related by a spectral flow term,
defined as follows. Let $(\scale dt)^2+\met_t$ denote the metric on
the cylinder $\R\times\Sigma$ induced by the family $\met_t$. By
restriction, the spin structures $\spinc_0$ and $\spinc_1$ on $Y$ and
$Y_1$ respectively induce spin structures on $Y-K$.  These can be
viewed as different $\SpinC$-connections on the same $\SpinC$
structure on $Y-K$, which we will connect by a path
$\{A_t\}$ of reducible connections. For definiteness,
we choose a path whose holonomies around the meridian are monotone
increasing from $0$ to $\OneHalf$.  By restricting to the cylinder
$\R\times\Sigma$, we get a path of reducibles $\{A_t\}$,
and hence a corresponding path
of Dirac operators $\Dirac_{A_t|\R\times\Sigma}$.  As we have seen
(Proposition~\ref{prop:SFCyl}), if the metric on $\R\times\Sigma$ is
sufficiently stretched out, then this spectral flow between the Dirac
operators is independent of the scale $\scale$. We denote this
spectral flow by $\SF_{\R\times\Sigma}(\spinc_0,\spinc_1)$.  With this
spectral flow defined, we turn our attention to the following key step
towards establishing Theorem~\ref{thm:SurgeryFormula}:

\begin{prop}
\label{prop:SurgerTheta}
Fix a generic path of metrics $\met_t$ so that $\rho(\ModSp_{\met_t}(Y-K))$
misses the spin structures $0\times 0$ and $\OneHalf\times 0$.  The
theta invariant satisfies:
\begin{equation}
\label{eq:SurgerTheta}
\theta_{\met_t}(Y_1)-\theta_{\met_t}(Y)=\left(\sum_{\spinc\in\SpinC(Y_0)}\theta(Y_0,\spinc)\right)
+ \SF_{\R\times\Sigma}(\spinc_0,\spinc_1).
\end{equation}
\end{prop}

\begin{proof}
Fix a real number $\delta$ with $0<\delta<\OneHalf-\epsilon$.  Fix
curves in $\Torus{2}$: $\gamma= 0\times S^1$, $\gamma_0=S^1\times
\{\delta\}$, and $\gamma_1=\{(s,s+\OneHalf)\}$. Note that the restriction of $\spinc_0$ and $\spinc_1$
to the torus gives the spin structures corresponding to $0\times 0$
and $\OneHalf\times 0$ respectively.  Since the moduli space misses
these spin structures, it follows that $\theta_{\met_t}(Y)$ and
$\theta_{\met_t}(Y_1)$ are well-defined, and indeed by the definition
of the $\theta$-invariant, we have that
$\#\rho^{-1}(\gamma)=\theta_{\met_t}(Y)$ and
$\#\rho^{-1}(\gamma_1)=\theta_{\met_t}(Y_1)$; similarly,
$$\#\rho^{-1}(\gamma_0)=
\left(\sum_{\spinc\in\SpinC(Y_0)}\theta(Y_0,\spinc)\right).$$

Consider the oriented subset $\Dom\subset \Torus{2}$ which does not
contain $\OneHalf\times\OneHalf$, and whose boundary is $\gamma_0 +
\gamma - \gamma_1$. Then, by transversality,
\begin{equation}
\label{eq:BoundModSp}
\partial \left(\rho^{-1}(\Dom)\right)=
\rho^{-1}(\gamma_0)+\rho^{-1}(\gamma)-\rho^{-1}(\gamma_1)+
\left(\partial\ModSp_{\met_t}(Y-K)\right)\cap \rho^{-1}(\Dom).
\end{equation}
According to Proposition~\ref{prop:StructureModSpace}, and our choice
of $\Dom$,
$\left(\partial\ModSp_{\met_t}(Y-K)\right)\cap \rho^{-1}(\Dom)$ consists of
boundary components where $s=t$. In fact,
Proposition~\ref{prop:SFCylBig} shows that
$$\#\left(\partial\ModSp_{\met_t}(Y-K)\right)\cap
\rho^{-1}(\Dom)=\SF_{\R\times\Sigma}(\spinc_0,\spinc_1),$$
so that counting points in Equation~\eqref{eq:BoundModSp}, we obtain
Equation~\eqref{eq:SurgerTheta}.
\end{proof}

We can understand the spectral flow on the cylinder in terms of data
on the knot complement, thanks to the splitting principle for spectral
flow. Specifically, we can connect $\spinc_0$ and $\spinc_1$ through
reducibles on the knot complement, endowed with a metric which is
$\Umet_0$ on $U_0$, $\met_t$ on the cylinder, and a fixed metric on
$U_1-\nbd{K}$ which is product-like near both the $\Sigma$-boundary
(where it is isometric to a collar around $\met_1$), and the torus
boundary of $\nbd{K}$. (In fact, for concreteness, we assume that it
has the form $S^1\times S^1$ near the knot, where we take the product
of longitude and meridian.)  According to the splitting principle,
then,
\begin{equation}
\label{eq:SplitKnotK}
\SF_{Y-K}(\spinc_0,\spinc_1)=\SF_{U_0}(\spinc_0,\spinc_1)+\SF_{\R\times\Sigma}(\spinc_0,\spinc_1)
+ \SF_{U_1-K}(\spinc_0,\spinc_1).
\end{equation}

Strictly speaking, in order to achieve the apropriate transversality,
so that the Dirac operator at the endpoints $\spinc_0$ and $\spinc_1$
have no kernel, we have relaxed the reducibility hypothesis, to include
perturbations of reducibles by one-forms which are compactly supported
in the two handlebodies (see the proof of Lemma~\ref{lemma:UDirac}).
Thus, for example, Equation~\eqref{eq:SplitKnotK} should be
interpreted as follows: fix a pair $a_0$ and $a_1$ of one-forms which
are compactly supported inside $U_0$ and $U_1-K$ respectively.
$\SF_{Y-K}(\spinc_0,\spinc_1)$ is the spectral flow of the family $A_t
+ a_0 + a_1$, where $A_t$ is the family of reducibles connecting
$\spinc_0$ to $\spinc_1$, and the spectral flows over $U_0$ and
$U_1-K$ are actually the restrictions of $A_t+a_0+a_1$ to these two
submanifolds.  In the interest of clarity, we suppress these
perturbations from the subsequent discussion: when we discuss to
``reducibles'', we mean, in fact, connections obtained by
perturbing reducibles in this manner. (Note also that these
perturbations leave the holonomies around attaching circles
unchanged.)

To relate the spectral flow on the knot complement with data on closed
manifolds, we have the following result, which could be called a
surgery formula for $\Corr$ (compare~\cite{RubermanMeyerhoff}, see
also~\cite{Lim}):

\begin{lemma}
\label{lemma:SurgerCorr}
Fix a cylindrical-end metric $\Kmet$ on $Y-K$, and let $\metU_0$,
$\metU_1$ be metrics on the genus one handlebody $U$ which agree with
$\Kmet$ along the boundary.  Then, for all sufficiently large $T$,
$$\Corr_{Y_1}(\Kmet\cup_{T}\metU_1)-\Corr_{Y}(\Kmet\cup_{T}\metU_0)=
\SF_{Y-K}(\spinc_0,\spinc_1) + \CorrU(k_1, \spinc_1)-\CorrU(k_0,
\spinc_0).$$
\end{lemma}

\begin{proof}
Recall that there is a natural cobordism $W$ from $Y$ to $Y_1$.  This
cobordism is obtained from $Y\times [0,1]$, and then attaching a
two-handle with $+1$ framing.  We have a natural inclusion
$$(Y-K)\times [0,1]\subset W,$$ which maps $(Y-K)\times \{0,1\}$ to
the boundary of $W$ (i.e. taking $(Y-K)\times\{0\}$ to the knot
complement, as a subset of $Y$, and $(Y-K)\times\{1\}$ to the
corresponding subset of $Y_1$). Fix a metric on $W$ whose restriction
to the region
$(Y-K)\times[0,1]$ is a product metric, its restriction to the $Y$
boundary agrees with $\Kmet\cup_T k_0$, and its restriction to $Y_1$
agrees with $\Kmet\cup_T k_1$.

Consider the $\SpinC$ structure $\spincX$ on $W$ whose first Chern
class generates $H^2(W;\Z)$. Endow $W$ with a $\SpinC$ connection in
$\spincX$ whose restriction to $(Y-K)\times [0,1]$ is a path of
reducibles, and whose first Chern form is compactly supported away
from the boundary (and hence it interpolates between $\spinc_0$ and
$\spinc_1$). By definition,
\begin{equation}
\label{eq:DifferenceCorrY}
\Corr_{Y_{1}}-\Corr_{Y} = \ind \Dirac_{A} -
\left(\frac{c_1(\spincX)^2-\sigma(W)}{8}\right)=\ind\Dirac_{A}.
\end{equation}

The index is calculated by an excision principle. Let
$Z=W-(Y-K)\times[0,1]$.  According to~\cite{MullerCorners}, the Dirac
operator on both $Z$ and $(Y-K)\times[0,1]$ is a Fredholm operator,
since the Dirac operator has no kernels on the ``corners'' $S^1\times
S^1\times\{0,1\}$ and the various boundaries $Y-K$ (coupled to
$\spinc_0$ and $\spinc_1$ respectively) and $[0,1]\times \Sigma_1$
(coupled to a sufficiently slowly-moving one-parameter family of
connections on $\Sigma_1$ which bound -- see
Proposition~\ref{prop:StretchOutKernel}).  Thus, the index splits as
$$\ind\Dirac_A = \ind {\Dirac_A}|_{(Y-K)\times[0,1]} +
\ind{\Dirac_A}|_{Z}.$$
The first term is the spectral flow $\SF_{Y-K}(\spinc_0,\spinc_1)$. To
understand the second term, we replace the knot complement $Y-K$ by a
much simpler knot-complement $\CDisk\times S^1$, endowed with a family
of connections whose holonomy around the $S^1$ factor goes from $0$ to
$\OneHalf$. Note that the manifold $W_0=\left(\CDisk\times
S^1\right)\cup_{S^1\times S^1} Z$ is a cobordism from $S^3$ to $S^3$:
indeed, it is $\CP{2}$ punctured at two points. Moreover, the connection $A|_Z$ extends over $W_0$, to give a connection $A_0$. Now, by the same splitting
principle,
$$\ind\Dirac_{A_0} = \ind {\Dirac_{A_0}}|_{(\CDisk\times S^1)\times[0,1]} +
\ind{\Dirac_{A_0}}|_{Z}.$$
The first term on the right hand side is a spectral flow for the Dirac
operator through flat connections on the manifold $\CDisk\times S^1$,
which has non-negative sectional curvatures; thus, the index vanishes. The
second term on the right is, of course, the same as
$\ind{\Dirac_A}|_{Z}$; thus, we have that
\begin{equation}
\label{eq:Index}
\ind\Dirac_{A_0}=\ind {\Dirac_A}|_Z
\end{equation}
Finally, by the same reasoning which gave
Equation~\eqref{eq:DifferenceCorrY}, we have that
\begin{equation}
\label{eq:DiffCorrS}
\ind{\Dirac_{A_0}}=\Corr((\CDisk\times S^1)\cup
(U,k_0))-\Corr((\CDisk\times S^1)\cup (U,k_1)),
\end{equation}
which is a difference of the APS invariant for the three-sphere
$S^3$. Now, according to Proposition~\ref{prop:CompareCorrections}
(together with the fact that $\CorrU(\CDisk\times S^1)\equiv 0$ for
any $\SpinC$ connections with traceless curvature), we have that
\begin{equation}
\label{eq:SplitCorrS}
\Corr((\CDisk\times S^1)\cup U_1)
-
\Corr((\CDisk\times S^1)\cup U_0)
=
\CorrU(U_0)-\CorrU(U_1).
\end{equation}
Combining Equations~\eqref{eq:DifferenceCorrY}-\eqref{eq:SplitCorrS},
we have established the lemma.
\end{proof}

The surgery formula for $\NInv$ now is an easy consequence of the
surgery formula for $\theta$ and $\Corr$:

\vskip.2cm
\noindent{\bf Proof of Theorem~\ref{thm:SurgeryFormula}.}
First we claim that the chambered properties of $\CorrU$ and $\Corr$,
the splitting formula (the version stated in
Equation~\ref{eq:SplitKnotK}), and Lemma~\ref{lemma:SurgerCorr}, we
see that
$$
\Big(\CorrY(Y_1,\spinc_1)-\CorrU(U_0,\spinc_1)-\CorrU(U_1',\spinc_1)\Big)
-
\Big(\CorrY(Y,\spinc_0)-\CorrU(U_0,\spinc_0)-\CorrU(U_1,\spinc_0)\Big)
= \SF_{\R\times
\Sigma}(\spinc_0,\spinc_1).
$$ It follows that
$$\NInv(Y_1)-\NInv(Y)=\Inv_{\met_t}(Y_1)-\Inv_{\met_t}(Y) -
\SF_{\R\times \Sigma}(\spinc_0,\spinc_1).$$
Proposition~\ref{prop:SurgerTheta} then finishes the proof.
\qed

config: arxiv


TITLE: Meaning of integral in equation
QUESTION [0 upvotes]: I was wondering that if $\int_0^xv(t)dt+C=\int v(x)dx$ where v is the velocity function and x is time, what is the meaning of $\int_0^xv(t)dt$ in the context of velocity or position or change in position? Does it even have any meaning because since $\int_0^xv(t)dt=\int v(x)dx-C$ and since C is unknown and $\int v(x)dx$ is an indefinite integral, I don't think $\int_0^xv(t)dt$ has meaning.

REPLY [0 votes]: The (definite) integral of the velocity between two times (in your case $0$ and $x$) is the (net) distance traversed in that time interval.
If you measure the position as the net distance from some fixed position you have a function whose derivative is the velocity. In that sense the position is an indefinite integral of the velocity. The "arbitrary constant" is there because the fixed starting position is arbitrary.


text:
\begin{document}

\title{The optimal trajectory to control complex networks}
\author{Aming Li$^{1,}$\footnote{amingli2011@gmail.com},
Long Wang$^{2,}$\footnote{longwang@pku.edu.cn},
and Frank Schweitzer$^{1,}$\footnote{fschweitzer@ethz.ch}}

\affiliation{
\begin{enumerate}
  \item Chair of Systems Design, Department of Management,
Technology and Economics, ETH Z\"urich, Weinbergstrasse 56/58, Z\"urich CH-8092, Switzerland
  \item Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China
\end{enumerate}
}
\date{\today}

\begin{abstract}
Controllability, a basic property of various networked systems, has gained profound theoretical applications in complex social, technological, biological, and brain networks.
Yet, little attention has been given to the control trajectory (route), along which a controllable system can be controlled from any initial to any final state, hampering the implementation of practical control.
Here we systematically uncover the fundamental relations between control trajectory and several other key factors, such as the control distance between initial and final states ($\delta$), number of driver nodes, and the control time.
The length ($\Lc$) and maximum distance to the initial state ($\Rc$) are employed to quantify the locality and globality of control trajectories.
We analyze how the scaling behavior of the averaged $\Lc$ and $\Rc$ changes with increasing $\delta$ for different initial states.
After showing the scaling behavior for each trajectory, we also provide the distributions of $\Lc$ and $\Rc$.
Further attention is given to the control time $t_{f}$ and its influence on $\Lc$ and $\Rc$.
Our results provide comprehensive insights in understanding control trajectories for complex networks, and pave the way to achieve practical control in various real systems.
\end{abstract}

\maketitle

\newpage
\section{Introduction}
As a powerful framework, complex networks have been widely employed to understand various complex systems,
where nodes indicate system's components and links capture interactions between them \cite{SocialNet94book,Havlin2004book,Barabasi04,barrat2008dynamical,Rajapakse2011pnas,Coyte2015Sci}.
Controllability---a basic property detecting whether a system can be controlled from external inputs
\cite{Liu2011,Wang2013,Gao2014,Posfai2014NJP,LiXiang2014,Liu2016Rev,Cornelius2013NatCommun,Chen2014,Chen2017,Li2017,Gu08Con,ConBrain2016Simu,Yan2017Nature,Bassett18NP}, helps to
uncover the principles of, for example, the interactions of neural circuits of cognitive function in brain networks \cite{Gu08Con}, or even predicting neuron function in the nematode \textit{Caenorhabditis elegans} \cite{Yan2017Nature}.
Indeed, a system is said to be controllable, if it can be driven from arbitrary initial state to arbitrary final state within finite time under appropriate control inputs  \cite{Kalman63,Xie2002tac,Chen2017}.
However, the reported principles of control cannot tell how systems behave under control inputs, namely, no information can be obtained on the evolution of a system's state in order to reach the desired state by just testing the system's controllability.
Although some results emerge on control cost (energy)
\cite{Rajapakse2011pnas,Yan2012PRL,Yan2015a,Lai2016,Klickstein2017,Bassett18NP}, the practical control trajectory (route) from the initial to final state along which the system must traverse all transient states, is far from understood, which strongly inhibits the practical applications.

Here we systematically explore control trajectories for controlling complex networks, revealing the fundamental relations between practical trajectories and control distance, number of driver nodes, and the control time.
Our findings clarify the fundamental behavior of practical control trajectories when we control complex networks, impulsing the real applications of the network control theory.

\section{Dynamics on Complex Networks}
The dynamics of a complex network with external inputs can be described mathematically as
\begin{equation}
\label{nonlineardy}
\dot{\mathcal{X}}(t) = \boldsymbol{f}(t, \mathcal{X} (t), \mathcal{U}(t), \mathcal{P}),
\end{equation}
where $\mathcal{X}_i(t)$ is the state of node $i$ at time $t$, like the level of neural activity of brain region $i$ in a brain network \cite{Gu08Con,Bassett18NP,Yan2017Nature},
or the concentration of metabolite $i$ in a metabolic network \cite{Almaas2004}.
The vector $\mathcal{X}(t)$ collects the state of all the $N$ nodes, \ie, $\mathcal{X}(t) = (\mathcal{X}_1(t),\mathcal{X}_2(t),\cdots,\mathcal{X}_N(t))^\textrm{T} \in\mathbb{R}^N$, represents the system state at time $t$.
$\boldsymbol{f}(*) = (f_{1}(*),f_{2}(*),\cdots,f_{N}(*))^\textrm{T}$ denotes interaction dynamics among nodes.
$\mathcal{U}(t)\in\mathbb{R}^p$ captures the input signals acting directly on $p$ ($\leq N$) nodes (namely, driver nodes \cite{Liu2011}).
$\mathcal{P}$ is the set of the system's parameters, which reflects the exact intensity that nodes interact with each other.

Due to the lack of empirical information about the
exact nonlinearity of $\boldsymbol{f}(*)$ and the related set of precise parameters $\mathcal{P}$, equation (\ref{nonlineardy}) is normally linearized to pursue analytical insights \cite{Liu2011,Gao16Nature,Yan2017Nature,Gu2017Traj,Bassett18NP}.
By assuming that the fixed point of the network is $\mathcal{X}^*$ without additional inputs, \ie, $\boldsymbol{f}(t, \mathcal{X}^*, \mathcal{U}^*)=0$, we linearize (\ref{nonlineardy}) by employing $\x(t) = \mathcal{X}(t) - \mathcal{X}^*$ and $\u(t) = \mathcal{U}(t)-\mathcal{U}^*$, arriving at the following dynamics
\begin{equation}
\label{lineardy}
\dot{\x }(t) = \A \x(t) + \B\u(t),
\end{equation}
in the time interval $[t_0, t_f]$ (see Fig.~\ref{fig_1_cartoon}a).
$\A= \frac{\partial \boldsymbol{f}(*)}{\partial  \mathcal{X}} \Big |_{\mathcal{X}^*, \mathcal{U}^*}$
corresponds to the adjacency matrix of the network (see Fig.~\ref{fig_1_cartoon}b and c), whose entry $a_{ij}$ represents, for example, the number of white matter streamlines linking from regions $j$ to $i$ in the brain network \cite{Gu2017Traj,Bassett18NP}.
$\B = \frac{\partial  \boldsymbol{f}(*)}{\partial  \mathcal{U} }\Big |_{\mathcal{X}^*, \mathcal{U}^*}$ gives the constant mapping between inputs and driver nodes of the network (see Fig.~\ref{fig_1_cartoon}a).

\section{Variables to quantify the control trajectory}

To quantify the control trajectory when we control complex networks, we adopt two variables.
One is the length
\begin{equation}
\label{Ldefmain}
\Lc = \int_{t_0}^{t_f}\| \dot{\x }(t)\| \mathrm{d}t
= \int_{t_0}^{t_f} \sqrt{ \dot{\x }^{\textrm{T}}(t) \dot{\x }(t) }\mathrm{d}t
\end{equation}
telling how long the control trajectory wanders in the controllable space.
Indeed, the length of control trajectory is widely used to quantify the locality of control trajectories for complex networks \cite{Sun2013prl,Li2017}, and it is also applied to analyze brain networks \cite{Gu2017Traj}.
It is discovered that $\Lc$ can be extremely large when the control distance $\delta = \|\x_f - \x_0 \|$ approaches $0$ \cite{Sun2013prl,Li2017}.
This implies that the optimal trajectory is probably nonlocal where in some dimensions the state components of the trajectory pass through highly extreme values (see Fig.~\ref{fig_1_cartoon}d).
Nevertheless, when $\Lc$ is large, it does not necessarily mean that the optimal trajectory is nonlocal.
Indeed, when the trajectory circuits around the initial state before arriving at the final one, $\Lc$ can still be large but the system state does not wander far from the initial state (see Fig.~\ref{fig_1_cartoon}d).
It means that the optimal trajectory cannot be solely reflected by the magnitude of $\Lc$.
Here we propose the radius of the control trajectory
\begin{eqnarray}
\label{DefDmain}
\Rc &=& \max_{t_0 \leq t \leq t_f} \| \x(t) - \x_0 \|
=
\max_{t_0 \leq t \leq t_f} \sqrt{ \sum_{i=1}^N \left(x_i(t) - x_i(t_0) \right)^2}
\end{eqnarray}
to quantify the maximum distance that the control trajectory deviates from the initial state among all of the system's intermediate states.
Here $\Rc$ can serve as a signal to dictate the existence of extreme values of state components.
Indeed, if there are some extremely large values of $x_i(t)$, then $\Rc$ will be large as well, and if the control trajectory is direct from the initial to the final state, then we have $\Rc \approx \delta$.

\section{The optimal control trajectory}

For the dynamics given in equation~(\ref{lineardy}), we obtain that, starting from $\x_0$ at time $t_0$, the control trajectory at the time $t$ $(<t_f)$ is
\begin{eqnarray}
\x (t)  =  \textrm{e}^{\A (t-t_0)}\x _0 + \int^t_{t_0} \textrm{e}^{\A (t-\tau)}\B \u(\tau) \textrm{d} \tau,
\end{eqnarray}
with the external input $\u(\tau)$.
To drive the network to reach the final state $\x_f$ at time $t_f$, however, we can choose an enormous number of different inputs (Fig.~\ref{fig_1_cartoon}d), which in turn generate different control trajectories with different control costs.
Indeed, the input control cost is defined as $E = \int^{t_f}_{t_0} \u(t)^{\text{T}} \u(t) \text{d} t$ \cite{OptimalBooLewis}, which reaches its minimum with the optimal control input
 $$\u(t)=\B^{\textrm{T}}\e^{\A ^\textrm{T}(t_{f}-t)}\W^{-1}[t_0, t_f] \textbf{\textrm{d}} $$
where
$\textbf{\textrm{d}} = \x_f - \e^{\A (t_f-t_0)}\x_0 $ is the difference between the desired final state $\x_f$ and the natural final state that the system evolves without external inputs, and
$\W = \int^{t_f}_{t_0} \e^{\A (t_f-\tau)}\B \B^{\textrm{T}}\e^{\A ^\textrm{T}(t_{f}-\tau)}  \textrm{d} \tau.$
Here, for given initial and final states, we focus on the optimal control trajectory determined by the optimal control inputs, along which the control cost is minimum.

\section{How initial states and control distances affect the averaged length (radius) of control trajectories}

When the network is steered from $\x_0$ to $\x_f$ in practice, it is of great interest how the direct control distance $\delta = \|\x_f - \x_0 \|$ affects the way from $\x_0$ to $\x_f$.
The length of the optimal control trajectory is
\begin{eqnarray}
\Lc =
 \int_{t_0}^{t_f}  \sqrt{
\| \x_0\|^2f(\bar{\x}_0,\bar{\x}_0) + 2\| \x_0\| \| \x_f\| f(\bar{\x}_0,\bar{\x}_f) + \| \x_f\|^2f(\bar{\x}_f,\bar{\x}_f)
}  \mathrm{d}t,
\end{eqnarray}
where $\bar{\x}_0$ and $\bar{\x}_f$ is the unit vector along the direction of $\x_0$ and $\x_f$ separately, and the function $f(*)$ is given in the Ref.~\cite{SM}.

The final state $\x_f = \x_0 + \delta \bar{\x}$ when $\bar{\x}$ is the unit vector along the direction of $\x_f - \x_0$.
This suggests that the behavior of control trajectories is determined by the relation between the initial state and the control distance.
Here we first focus on the overall behavior of the averaged length ($\Lc$) of control trajectories under the same direct control distance as a function of $\delta$.

When $\x_0 = \zero$ ($\|\x_f\| = \delta$), we know that
$\Lc(\zero, l \x_f)
=
l \Lc(\zero, \x_f)$.
That means, when a network is controlled from the origin, the averaged length of the control trajectory increases linearly with the control distance, \ie, $\Lc \sim \delta$ (see Fig.~\ref{fig_L_delta}a).

When $\x_0 \neq \zero$, from $\x_f = \x_0 + \delta \bar{\x}$, we find that:
(i) With the increase of $\delta$ (say, bigger than the critical value $\delta^*$), the effect of $\x_0$ can be neglected, leading to  $\Lc(\x_0, l\x_f)  \approx l \Lc(\zero, \x_f)$,
which follows the laws of the scenario for $\x_0 = \zero$.
That is to say, when the control distance is relatively long compared to the norm of the initial state, it will dominate the scaling behavior of the averaged length of control trajectories (see Fig.~\ref{fig_L_delta}a);
(ii) When the control distance is relatively short ($\delta<\delta^*$) with a nonzero initial state, we find that the averaged $\Lc$ can be approximated by the constant
\begin{eqnarray}
\label{Lstarx0}
\Lc^*=
\|\x_0 \| \int_{t_0}^{t_f}  \sqrt{
f(\bar{\x}_0,\bar{\x}_0) + 2 f(\bar{\x}_0,\bar{\x}_f) + f(\bar{\x}_f,\bar{\x}_f)
}  \mathrm{d}t,
\end{eqnarray}
This means that the averaged length of the control trajectory is dominated by $\| \x_0 \|$ as a constant when the control distance is short (see Fig.~\ref{fig_L_delta}a).

Equation (\ref{Lstarx0}) also tells us that the averaged constant increases linearly with the norm of the initial state, \ie, $\Lc^* \sim \| \x_0 \|$ since $\Lc^*(l\x_0,\x_f) = l\Lc^*(\x_0,\x_f)$ (see Fig.~\ref{fig_L_delta}c).

As to the critical value of $\delta$ at which the behavior of the averaged $\Lc$ will alter, we know that when $\x_0 = \zero$, $\Lc = k_1 \delta$, and when $\x_0 \neq \zero$, the corresponding constant is $k_2 \| \x_0 \|$, both $k_1$ and $k_2$ are constants.
Thus, at the critical control distance $\delta^*$,  we have $k_1 \delta^* = k_2 \| \x_0 \|$, meaning that the scaling behavior of $\delta^*$ follows $\delta^* \sim  \| \x_0 \|$.
This is also validated with numerical calculations (see Fig.~\ref{fig_L_delta}b).

Taken together, we find an universal linear scaling behavior of both the averaged length and averaged radius of the optimal control trajectory, namely, $\Lc \left(\Rc\right) \sim \delta$, $\delta^* \sim  \| \x_0 \|$, and $\Lc^* \left(\Rc^*\right) \sim \| \x_0 \|$.

\section{Scaling behavior of each control trajectory and its distribution}
The averaged values of $\Lc$ and $\Rc$ provide statistical insights of control trajectories at the same control distance.
In the phase space, however, for two opposite final states ($\x_{f1}$ and $\x_{f3}$ in Fig.~\ref{fig_dist_LR}a), when their control distances to a given initial state ($\x_0$ in Fig.~\ref{fig_dist_LR}) are equal, they can correspond to totally different control objectives.
Indeed, for neural activity ($x_i(t_f)$) of the brain region $i$, the two final states $x_i(t_f)=1$ and $0$ have the same distance to the initial state $x_i(t_f)=0.5$, but $1$ and $0$ capture totally opposite states.
Thus, simply averaging over $\Lc$ or $\Rc$ for trajectories with the same $\delta$ may probably miss out the potential fundamental laws behind the practical control routes.
To better understand this, we first focus on each separate trajectory and then explore the statistical characteristics of all trajectories.

Interestingly, we find that for nonzero initial state, $\Lc$ has the inverse scaling behavior for the opposite final states with the same small $\delta$.
For example, when $\x_0 = \tilde{\x}_0 \neq \zero$ (Fig.~\ref{fig_dist_LR}a), we randomly select a final state ($\x_f = \tilde{\x}_{f1}$) with direct distance $\delta$ to $\tilde{\x}_0$.
We find that the corresponding length of control trajectory first decreases with $\Lc = -a\delta + b$ and then increases linearly with $\Lc = a\delta - b$ (solid upward-pointing triangle in Fig.~\ref{fig_dist_LR}b).
As to the opposite direction ($\x_f = \tilde{\x}_{f3}$), we have $\Lc = a\delta + b$ (solid downward-pointing triangle in Fig.~\ref{fig_dist_LR}b).
When we average $\Lc$ over the final states with $\tilde{\x}_{f1}$ and $\tilde{\x}_{f3}$, we find that $\Lc$ first stays constant and then shares the same scaling law as for $\x_0 = \zero$ (grey solid square in Fig.~\ref{fig_dist_LR}b), which is in line with the results reported in Fig.~\ref{fig_L_delta}.
Thus the averaged $\Lc$ over different control trajectories with same control distance neutralizes the inverse scaling behavior for opposite final sates.

For $\x_0 = \zero$ (Fig.~\ref{fig_dist_LR}a) and $\x_f = \x_{f2}$, we have $\Lc = a\delta$ (green solid circle in Fig.~\ref{fig_dist_LR}b), and $\Lc \sim \delta$ holds for any specific final state \cite{SM}.

In addition, as to any pair of opposite final states, the lengths of control trajectories are equal \cite{SM}.

Furthermore, for all the control trajectories at the same control distance (Fig.~\ref{fig_dist_LR}c), the cumulative distribution function of $\Lc$ is
\begin{eqnarray}
P (\Lc \leq x) = \frac{2}{\pi}  \arcsin \frac{x}{2r},
\label{mainadfsi}
\end{eqnarray}
where $2r$ is the maximum value of $\Lc$.
The above function can predict the numerical results very well (Fig.~\ref{fig_dist_LR}d).

When $\x_0 \neq \zero$, the constant nonzero initial state determines the uniform distribution of $\Lc$ for small $\delta$, while for large $\delta$, $\Lc$ has the same distribution given by the above equation for both zero and nonzero initial states.
All the above results are applicable for the radius of control trajectories ($\Rc$), and other more results are given in the Ref.~\cite{SM}.

\section{How control time affects the scaling behavior of $\Lc$ and $\Rc$}
Under a given control distance, the control time ($t_f-t_0$) that control signals can harness to drive the system to the final state is quite important.
It affects not only the velocity of system state change but also the corresponding minimal control energy.
Here we seek to address how the control time affects the scaling behavior of $\Lc$ and $\Rc$.
According to equation (\ref{Ldefmain}), we have
\begin{eqnarray}
\label{Lbasicfort}
\Lc
=
\int_{0}^{t_f} \sqrt{
\x_f ^\text{T} \W^{-1}[0, t_f] \textrm{e}^{\A(t_{f}-t)} \left( \W[0,t]\A  + \mathbf{I} \right)
\left( \A \W[0,t] + \mathbf{I} \right)
\textrm{e}^{\A ^\textrm{T}(t_{f}-t)}
\W^{-1}[0, t_f] \x_f
 }\mathrm{d}t.
 \end{eqnarray}
To theoretically analyze the relation between $\Lc$ and the control time $t_f$, we divide it into three situations according to the number of driver nodes, \ie, one driver node, $p~(1 < p <N)$ driver nodes, and $N$ driver nodes \cite{SM}.

Note that, without loss of generality, here we set $\x_0 = \zero$ and $t_0 = 0$.
We numerically show the results as follows.

For one driver node and short control time $t_f$, we find that $\Lc$ ($\Rc$) decreases with the power-law function of $t_f$ when the system is asymptotically stable (the maximum eigenvalue $\lambda_1$ of $\A$ is smaller than $0$) or unstable ($\lambda_1>0$) (Fig.~\ref{fig4_N5_LR_t}a and~\ref{fig4_N5_LR_t}c).
With the increase of $t_f$ for an asymptotically stable system, $\Lc$ ($\Rc$) will first keep as a constant and then decrease again with the same power-law function, and eventually keep as the constant $\alpha$ for big $t_f$ (Fig.~\ref{fig4_N5_LR_t}a).
We find that $\alpha \approx \delta$ and $\Lc \approx \Rc$ if $t_f$ is big, meaning that the control trajectory goes straight from the initial to the final state when the control time is long enough.

Interestingly, when the system is unstable, $\Lc$ ($\Rc$) keeps $\delta$ with the increase of $t_f$ (Fig.~\ref{fig4_N5_LR_t}c), while in this case we know that the minimum control energy is $E_{\min} \sim \text{e}^{- 2 \lambda_1 t_f}$ \cite{Yan2012PRL,Li2017ConEng}.
That is to say, for unstable systems, when more control \emph{time} is given, the corresponding optimal trajectory stays constant despite that the minimum energy needed to reach final state decreases exponentially.
For the critical scenario where $\lambda_1 = 0$, we find that $\Lc$ ($\Rc$) equals  $\delta$ irrespective of how much control time is given (Fig.~\ref{fig4_N5_LR_t}b).
This means that, although the control time is short, an increasing control time can reduce the control energy dramatically \cite{Yan2012PRL,Li2017ConEng}.
But neither the length nor the radius of the control trajectory can be secured.

By adding more driver nodes, both $\Lc$ and $\Rc$ decrease, and the exponent of the scaling behavior of $\Lc$ and $\Rc$ will decrease as well (Fig.~\ref{fig4_N5_LR_t}d-f).
When we control all nodes directly, \ie, when the number of driver nodes is equal to the system size, both the length and radius of control trajectories keep constant for different scenarios of stability of the system and control time (Fig.~\ref{fig4_N5_LR_t}g-i).

\section{Conclusion and Discussions}

We statistically analyze the averaged length and the averaged radius of control trajectories with the same control distance.
We also provide the scaling behavior of these two quantities.
We demonstrate that aggregating the length (radius) of trajectories over many evenly selected final states \emph{neutralizes} the embedded scaling behavior for each single trajectory.
For example, as $\x_0 = \zero$, the linear scaling behavior of $\Lc$ ($\Rc$) for every pair of opposite final states has the contrary sign for short control distance. Averaging this will make $\Lc$ ($\Rc$) a constant.
Thus the statistical results are not enough to fully understand the control trajectories.
Apart from uncovering the relations of the scaling for different final states equidistant to $\x_0$, we also analytically provide the distribution of $\Lc$ and $\Rc$.
In addition, $\Lc$ and $\Rc$ can be employed to classify different kinds of optimal control trajectory in terms of the locality and globality in various empirical systems.

Another key factor to implement control under practical circumstances is the control time ($t_f$), \ie~the time needed to reach the final state.
We find that for short $t_f$, $\Lc$ ($\Rc$) is a power law function of $t_f$, meaning that, in this case $\Lc$ ($\Rc$) can be dramatically reduced if slightly more time is given.
When $t_f$ is big, $\Lc$ ($\Rc$) cannot be affected too much either by increasing the number of driver nodes or by changing the stability of the system.
This has consequences \textit{e.g.} for cognitive control, where the brain can quickly achieve some complex cognitive functions by altering the dynamics of neural systems with energetic inputs \cite{CogCon02,CogCon13,CogCon14}.
Our findings suggest that the $\Lc$ ($\Rc$) of the optimal control trajectories in the phase space of neural activity can be largely conserved when more time is given to the brain to perform the cognitive control.

To pursue the analytical insights of optimal control trajectories, we linearize the general nonlinear system.
Indeed, linearization has become the norm in analyzing diverse networked systems \cite{Liu2011,Rajapakse2011pnas,Gao16Nature,Yan2017Nature,Gu2017Traj,Bassett18NP} due to several reasons.
One is that the empirical nonlinearity and the related parameters are hard to quantify and to estimate.
Another one is governed by a lemma that if the linearized system of a nonlinear dynamics is controllable along a specific trajectory, the nonlinear system is also controllable along the same trajectory \cite{LinearizationBook}.
The basic theoretical laws and insights of the practical control trajectory from initial to final state uncovered here facilitate the implementation of actual control in various empirical systems.
And it is worth further investigating for generalized scenarios of general nonlinear dynamics \cite{Khalil2002a} of static networks or temporal networks \cite{Holme2012,NaokiBook,Li2017,Li2017ConEng}.


TITLE: Height of a full binary tree
QUESTION [8 upvotes]: A full binary tree seems to be a binary tree in which every node is either a leaf or has 2 children.
I have been trying to prove that its height is O(logn) unsuccessfully.
Here is my work so far:
I am considering the worst case of a full binary tree in which each right node has a subtree, and each left node is a leaf.
In this case:
$N = 2x - 1$
$H = x - 1$
I am going nowhere trying to prove that $H = O(log(N))$
Furthermore, we know that leaves l is bounded by $h+1 <l<2^h$.
Internal nodes is bounded by $h<i<2^{h-1}$.
All this proves is that number of nodes $n=i+e$ is $<= 2^{h+1} - 1$ i.e. $log(n) <= h$. But this does not take me anywhere closer to prove that $H = O(log(n))$

REPLY [3 votes]: Given h...height if tree, N(h).. count of nodes for tree height h.
If
  h = 1: N(h) = 1;
  h = 2: N(h) = N(1) * 2 = 1 * 2;
  h = 3: N(h) = N(2) * 2 = N(1) * 2 * 2 = 1 * 2 * 2 * 2;
  ...
  h = n: N(n) = N(n-1) * 2 = ... = 1 * 2 * 2 * 2...= 1 * 2^n = 2^n
Each node has two children.
So, count of nodes if height of tree is n is N(n) = 2^n.
And if count of nodes of full binary tree is N, then height of tree n is proportional to
log_2(N) or n = C(log(N)).


TITLE: Linear conjugate of a cubic polynomial
QUESTION [2 upvotes]: In any polynomial ring (but especially $K[x]$, $K$ number field) Is it true that for any cubic polynomial $p(x)$ there exist a linear polynomial $l(x)$ such that $l\circ p\circ l^{-1}=x^3+c$ for some constant $c\in K$?
We know it exists for quadratic polynomials but i wonder it exists for cubics or not.
Thank you.

REPLY [1 votes]: If such an $l$ exists then $p(x) = a (x+b)^3 + c$, for some $a$, $b$, $c\in \mathbb{C}$, $a\ne 0$. That means $p'(x)$ has a double root $-b$. That is almost never the case.
However, it is possible to write $p$ as the sum of two cubes of linear functions.