# Duals of Tirilman spaces have unique subsymmetric basic sequences

S. J. Dilworth  Department of Mathematics, University of South Carolina,
Columbia, SC 29208, USA<EMAIL_ADDRESS>,  D. Kutzarova  Department of
Mathematics, University of Illinois Urbana-Champaign, Urbana, IL 61807, USA;
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences,
Sofia, Bulgaria<EMAIL_ADDRESS>,  B. Sarı  Department of Mathematics,
University of North Texas 1155 Union Circle 311430 Denton, Texas 76203-5017,
USA<EMAIL_ADDRESS>and  S. Stankov<EMAIL_ADDRESS>
###### Abstract.

The Tirilman spaces $Ti(p,\gamma)$, $1<p<\infty$, were introduced by Casazza
and Shura as variations of the spaces constructed by Tzafriri. We prove that
all subsymmetric basic sequences in the dual space $Ti^{*}(p,\gamma)$ are
equivalent to its canonical subsymmetic but not symmetric basis.

###### 2000 Mathematics Subject Classification:

46B03, 46B06, 46B10, 46B25

The first author was supported by Simons Foundation Collaboration Grant No.
849142. The second author was supported by Simons Foundation Collaboration
Grant No. 636954. The first three authors thank the Workshop in Analysis and
Probability at Texas A&M University (2022) for support where this work was
initiated.

##  1\. Introduction

Symmetric structures play an important role in the theory of Banach spaces. A
basic sequence $(x_{j})_{j=1}^{\infty}$ is symmetric if the rearranged
sequence $(x_{\pi(j)})_{j=1}^{\infty}$ is equivalent to
$(x_{j})_{j=1}^{\infty}$ for any permutation $\pi$ of $\mathbb{N}$. Recall
that a sequence $(x_{j})_{j=1}^{\infty}$ is a basic sequence if it is a
(Schauder) basis of its closed linear span; two basic sequences
$(x_{j})_{j=1}^{\infty}$ and $(y_{j})_{j=1}^{\infty}$ are said to be
equivalent provided a series $\sum_{j=1}^{\infty}a_{j}x_{j}$ converges if and
only if $\sum_{j=1}^{\infty}a_{j}y_{j}$ does.

The class of subsymmetric basic sequences, that is, those that are
unconditional and equivalent to all of their subsequences [LT], is formally
more general than the class of symmetric ones. For a while, these two concepts
were believed to be equivalent until Garling [G] provided a counterexample.
Later, subsymmetric bases became important on their own within the general
theory. For instance, the first arbitrarily distortable Schlumprecht space [S]
has a subsymmetric basis which is not symmetric.

Albiac, Ansorena and Wallis [AAW] used Garling-type spaces to provide the
first example of a Banach with a unique subsymmetric basis which is not
symmetric. However, as shown in a sequel paper [AADK], that space contains a
continuum of non-equivalent subsymmetric basic sequences. Altshuler [A] (see
also Example 3.b.10 in [LT]) constructed a space which is not isomorphic to
$c_{0}$ or $\ell_{p}$ for any $1<p<\infty$ and in which all symmetric basic
sequences are equivalent to its symmetric basis. Recently, the first example
of a Banach space with a unique subsymmetric basic sequence which is not
symmetric is given in [CDKM]. That answered a question posed in [KMP] and
[AADK]. The space under consideration was $Su(T^{*})$ [CS], the subsymmetric
version of $T^{*}$. As it became customary, $T$ is the space considered by
Figiel and Johnson [FJ] and its dual $T^{*}$ is the original space constructed
by Tsirelson [T], the first example of a space which does not contain an
isomorphic copy of $c_{0}$ or $\ell_{p}$, $1\leq p<\infty$.

In this paper we give more examples of spaces with a subsymmetric but not
symmetric basis which contain, up to equivalence, a unique subsymmetric basic
sequence. These examples are based on Tzafriri spaces. Tzafriri [Tz] had
constructed (counter)-examples of spaces with (symmetric bases) showing that
the notions of equal-norm type $p$ and equal-norm-cotype $q$ are not
equivalent to the notions of type $p$ and cotype $q$ for $p,q\neq 2$,
respectively. The Tirilman spaces $Ti(p,\gamma)$, where $1<p<\infty$ and
$0<\gamma<1$, are modified Tzafriri spaces, which were introduced and studied
by Casazza and Shura [CS]. They were named after Tzafriri’s Romanian surname.
We prove that for $1<p<\infty$ and sufficiently small $0<\gamma<1$, the dual
space $Ti^{*}(p,\gamma)$, whose canonical basis is subsymmetric but not
symmetric contains, up to equivalence, a unique subsymmetric basic sequence.
That is, all the subsymmetric basic sequences are equivalent to the canonical
basis. The method of our proof is parallel to the one in [CDKM]: While there
the normalized block bases $(x_{j})$ of the canonical basis of $Su(T^{*})$
with the property $\|x_{j}\|_{\infty}\to 0$ are shown to be asymptotic-$c_{0}$
sequences, we show that the similar block bases in $Ti^{*}(p,\gamma)$ yield
asymptotic-$\ell_{q}$ sequences, where $\frac{1}{p}+\frac{1}{q}=1$. Moreover,
unlike its dual $Ti(p,\gamma)$ has continuum many non-equivalent subsymmetric
basic sequences. This follows immediately from Theorem 21 of [CDKM] which
states that if a subsymmetric basis $(e_{i})$ is not equivalent to the unit
vector basis of $c_{0}$ or $\ell_{p}$ then either $(e_{i})$ or $(e^{*}_{i})$
admits a continuum of non-equivalent subsymmetric block bases.

##  2\. Spaces with a unique subsymmetric basic sequence

Given two basic sequences $(x_{n})_{n=1}^{\infty}$ and
$(y_{n})_{n=1}^{\infty}$ in Banach spaces $X$ and $Y$, respectively, we say
that $(x_{n})_{n=1}^{\infty}$ _$K$ -dominates_ $(y_{n})_{n=1}^{\infty}$ if the
bounded linear operator $T(x_{n})=y_{n}$ from $[(x_{n})_{n=1}^{\infty}]$ to
$[(y_{n})_{n=1}^{\infty}]$ has norm $\|T\|\leq K$. We say that
$(x_{n})_{n=1}^{\infty}$ _dominates_ $(y_{n})_{n=1}^{\infty}$ if
$(x_{n})_{n=1}^{\infty}$ $K$-dominates $(y_{n})_{n=1}^{\infty}$ for some
$K<\infty$. A _block basis_ with respect to a basic sequence
$(x_{n})_{n=1}^{\infty}$ is a sequence $(y_{n})_{n=1}^{\infty}$ of non-zero
vectors of the form $y_{n}=\sum_{k=p_{n}+1}^{p_{n+1}}a_{k}x_{k}$ where
$p_{1}<p_{2}<\cdots$ is an increasing sequence of natural numbers. For a
vector $x$ in the closed linear span of $(x_{n})_{n=1}^{\infty}$, its support
(with respect to $(x_{n})_{n=1}^{\infty}$) is the set of indices of its non-
zero coefficients. For finite sets of natural numbers $E$ and $F$ we say that
$E<F$ if $\max(E)<\min(F)$. For a natural number $n$, we say $n<x$, resp.
$n\leq x$, if $n<\min(\operatorname{supp}(x))$, resp.
$n\leq\min(\operatorname{supp}(x))$. A basic sequence $(x_{n})$ is called
_1-subsymmetric_ if it is $1$-unconditional and isometrically equivalent to
its subsequences.

A basic sequence $(x_{j})_{j=1}^{\infty}$ is called _(strongly) asymptotic_
-$\ell_{p}$, $1\leq p<\infty$ if there exist a constant $C>0$ such that for
every $m\in\mathbb{N}$ there is an $M\in\mathbb{N}$ such that for every
normalized block basis $(y_{j})_{j=1}^{m}$ of $(x_{j})_{j=M}^{\infty}$ and any
set of real numbers $(a_{i})$, we have

$\frac{1}{C}\left(\sum_{i=1}^{m}|a_{i}|^{p}\right)^{\frac{1}{p}}\leq\left\|\sum_{i=1}^{m}a_{i}y_{i}\right\|\leq
C\left(\sum_{i=1}^{m}|a_{i}|^{p}\right)^{\frac{1}{p}}.$

Although we will drop the term ‘strongly’ when referring to
asymptotic-$\ell_{p}$ sequences, it is important to note this is a stronger
version of the original definition from [MMT] which was given in a more
general setting.

Let $1<p<\infty$ and $0<\gamma<1$. As in the case of Tsirelson space, the norm
is defined via an implicit equation. For all $a=(a_{i})\in c_{00}$, the linear
space of finitely supported real-valued sequences, define

$\|a\|=\max\left\\{\|a\|_{\infty},\gamma\sup\frac{\sum_{j=1}^{n}\|E_{j}a\|}{n^{\frac{1}{q}}}\right\\},$

where $\frac{1}{p}+\frac{1}{q}=1$ and the inner supremum is taken over all
finite consecutive sets of natural numbers $1\leq E_{1}<\cdots<E_{n}$ and all
$n$. This norm can be computed via the limit of a recursive sequence of norms.
We refer to [CS], Section X.d.5, for more details. The Tirilman space
$Ti(p,\gamma)$ is the completion of $(c_{00},\|\cdot\|)$. It follows from the
definition that the unit vectors $(e_{n})_{n=1}^{\infty}$ form a
1-subsymmetric basis for $Ti(p,\gamma)$. We shall summarize some of their
known properties. The first one is the obvious analogue of Proposition X.d.8
[CS] which was proved for $Ti(2,\gamma)$.

######  Proposition 1.

For every $1<p<\infty$ and $0<\gamma<1$, the canonical basis
$(e_{n})_{n=1}^{\infty}$ is $1$-dominated by every normalized block basis of
$(e_{n})_{n=1}^{\infty}$.

Some further properties of $Ti(p,\gamma)$ that were proved in [CS] for
$Ti(2,\gamma)$ were listed in Theorem 6.1 [Sa].

######  Proposition 2.

Let $1<p<\infty$. Then for sufficiently small $0<\gamma<1$ the following hold
for $Ti(p,\gamma)$.

$(i)$ for any normalized successive blocks $(x_{j})_{j=1}^{\infty}$ of the
basis $(e_{i})$, we have

$\gamma n^{\frac{1}{p}}\leq\left\|\sum_{j=1}^{n}x_{j}\right\|\leq
3^{\frac{1}{q}}n^{\frac{1}{p}}.$

$(ii)$ $Ti(p,\gamma)$ does not contain isomorphs of any $\ell_{r}$, $1\leq
r<\infty$ or of $c_{0}$. In particular, $Ti(p,\gamma)$ is reflexive.

Remark. We shall apply the above proposition for $\gamma<3^{-\frac{1}{q}}$.

Actually, we need the more general version of the right-hand inequality of
$(i)$, which is the $p$-analogue of Lemma X.d.4 [CS].

######  Proposition 3.

If $0<\gamma<3^{-\frac{1}{q}}$ and $(x_{j})_{j=1}^{n}$ are block vectors in
$Ti(p,\gamma)$ with consecutive supports, $n\in\mathbb{N}$, then

$\left\|\sum_{j=1}^{n}x_{j}\right\|\leq
3^{\frac{1}{q}}\left(\sum_{j=1}^{n}\|x_{j}\|^{p}\right)^{\frac{1}{p}}.$

As an immediate corollary we obtain the following

######  Lemma 4.

Let $0<\gamma<3^{-\frac{1}{q}}$. Let $(x^{*}_{j})$ be a normalized block basis
of $(e_{j}^{*})$ in the dual space $Ti^{*}(\gamma,p)$. Then for every $n$ and
every choice of real numbers $(a_{j})_{j=1}^{n}$, we have

$\left\|\sum_{j=1}^{n}a_{j}x_{j}^{*}\right\|\geq\frac{1}{3^{\frac{1}{q}}}\left(\sum_{j=1}^{n}|a_{j}|^{q}\right)^{\frac{1}{q}}.$

###### Proof.

For any $1\leq j\leq n$ choose an $x_{j}\in Ti(\gamma,p)$ with $\|x_{j}\|=1$
and $x_{j}^{*}(x_{j})=1$. Let $(a_{j})_{j=1}^{n}$ be a set of real numbers. By
1-unconditionality we may assume that $a_{j}\geq 0$ and
$\operatorname{supp}x_{j}\subseteq\operatorname{supp}x^{*}_{j}$. Then by
duality,

$\displaystyle\sum_{j=1}^{n}a_{j}^{q}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}a_{j}x_{j}^{*}(a_{j}^{\frac{q}{p}}x_{j})=\left(\sum_{j=1}^{n}a_{j}x_{j}^{*}\right)\left(\sum_{j=1}^{n}a_{j}^{\frac{q}{p}}x_{j}\right)$
$\displaystyle\leq$
$\displaystyle\left\|\sum_{j=1}^{n}a_{j}x_{j}^{*}\right\|\left\|\sum_{j=1}^{n}a_{j}^{\frac{q}{p}}x_{j}\right\|$
$\displaystyle\leq$
$\displaystyle\left\|\sum_{j=1}^{n}a_{j}x_{j}^{*}\right\|3^{\frac{1}{q}}\left(\sum_{j=1}^{n}a_{j}^{q}\right)^{\frac{1}{p}},$

which gives the needed inequality. ∎

######  Proposition 5 ([Sa]).

Let $1<p<\infty$ and let $\gamma>0$ be sufficiently small. Then $Ti(p,\gamma)$
contains no symmetric basic sequence.

Remark. It was proved in [JKO] that $c_{0}$ is finitely representable in
$Ti(2,\frac{1}{2})$ (disjointly w.r.t. $(e_{j})$) which provides an
alternative proof that $(e_{j})$ is not symmetric.

######  Lemma 6.

Let $(e_{i})$ be a $1$-unconditional basis of a reflexive Banach space $X$
which is $K$-dominated by its normalized block bases, where $K\geq 1$. Then
$(e_{i}^{*})$ $K$-dominates all normalized block bases of $(e_{i}^{*})$ in the
dual space $X^{*}$.

###### Proof.

Let $(x_{i}^{*})$ be a normalized block-basis of $(e_{i}^{*})$ and let
$(a_{i})_{i=1}^{n}$, $n\in\mathbb{N}$, be an arbitrary set of real numbers.
$(e_{i}^{*})$ is also 1-unconditional, so we may assume that $a_{i}\geq 0$ for
all $1\leq i\leq n$. Pick a norming element $w\in X$, $\|w\|=1$,
$\left(\sum_{i=1}^{n}a_{i}x_{i}^{*}\right)(w)=\left\|\sum_{i=1}^{n}a_{i}x_{i}^{*}\right\|$.
Denote $A_{i}=\operatorname{supp}(x_{i}^{*})$.

The 1-unconditionality of $(e_{i})$ allows us to assume that

$\operatorname{supp}(w)\subseteq\bigcup_{i=1}^{n}A_{i}.$

Let $w_{i}=w|_{A_{i}}$ be the restriction of $w$ to the set $A_{i}$. Denote
$\|w_{i}\|=c_{i}$ and $B=\\{1\leq i\leq n\colon c_{i}\neq 0\\}$. By
1-unconditionality, $c_{i}\leq 1$, $1\leq i\leq n$. For each $i\in B$, let
$z_{i}=\frac{w_{i}}{c_{i}}$. Clearly $(z_{i})_{i=1}^{n}$ is a normalized
block-basis of $(e_{i})_{i=1}^{\infty}$ and

$w=\sum_{i\in B}c_{i}z_{i}.$

Then,

$\displaystyle\left\|\sum_{i=1}^{n}a_{i}x_{i}^{*}\right\|$ $\displaystyle=$
$\displaystyle\left(\sum_{i=1}^{n}a_{i}x_{i}^{*}\right)\left(\sum_{i\in
B}c_{i}z_{i}\right)$ $\displaystyle=$ $\displaystyle\sum_{i\in
B}a_{i}c_{i}x_{i}^{*}(z_{i})\leq\sum_{i\in B}a_{i}c_{i}$ $\displaystyle=$
$\displaystyle\left(\sum_{i\in B}a_{i}e_{i}^{*}\right)\left(\sum_{i\in
B}c_{i}e_{i}\right)\leq\left\|\sum_{i\in
B}a_{i}e_{i}^{*}\right\|\cdot\left\|\sum_{i\in B}c_{i}e_{i}\right\|$

By the $K$-domination,

$\left\|\sum_{i\in B}c_{i}e_{i}\right\|\leq K\left\|\sum_{i\in
B}c_{i}z_{i}\right\|=K.$

Thus,

$\left\|\sum_{i=1}^{n}a_{i}x_{i}^{*}\right\|\leq K\left\|\sum_{i\in
B}a_{i}e_{i}^{*}\right\|\leq K\left\|\sum_{i=1}^{n}a_{i}e_{i}^{*}\right\|.$

∎

######  Lemma 7.

For any $n$ and any sequence of normalized blocks $(x_{j}^{*})_{j=1}^{n}$ of
$(e_{j}^{*})_{j=1}^{\infty}$ in $Ti^{*}(p,\gamma)$,

$\left\|\sum_{j=1}^{n}x_{j}^{*}\right\|\leq\frac{n^{\frac{1}{q}}}{\gamma}.$

###### Proof.

By the previous Lemma 6 and Proposition 1, $(x_{j}^{*})_{j=1}^{n}$ is
1-dominated by $(e_{j}^{*})_{j=1}^{n}$, so

$\left\|\sum_{j=1}^{n}x_{j}^{*}\right\|\leq\left\|\sum_{j=1}^{n}e_{j}^{*}\right\|.$

The vector $\frac{\gamma}{n^{\frac{1}{q}}}\sum_{j=1}^{n}e_{j}^{*}$ belongs to
the unit ball of $Ti^{*}(p,\gamma)$, see e.g. [M], so
$\left\|\sum_{j=1}^{n}e_{j}^{*}\right\|\leq\frac{n^{\frac{1}{q}}}{\gamma}$. ∎

######  Lemma 8.

$Ti^{*}(p,\gamma)$ does not contain an isomorphic copy of $\ell_{q}$
$(\frac{1}{p}+\frac{1}{q}=1)$.

###### Proof.

Assume the contrary. Without loss of generality we may assume that a
normalized block basis $(x_{j}^{*})$ of $(e_{j}^{*})$ is $C$-equivalent to the
unit vector basis of $\ell_{q}$. Denote
$I_{j}=\operatorname{supp}(x_{j}^{*})$. Choose norming elements $x_{j}\in
Ti(p,\gamma)$, $\|x_{j}\|=1$, $x^{*}_{j}(x_{j})=1$. By the 1-unconditionality
we may assume that $\operatorname{supp}(x_{j})\subseteq
I_{j}\subset\mathbb{N}$ for all $j\in\mathbb{N}$. Clearly,
$I_{1}<I_{1}<\cdots$ and denote by $P_{j}$ the projection on $I_{j}$.

Define the projection

$P(x^{*})=\sum_{j=1}^{\infty}\langle P_{j}(x^{*}),x_{j}\rangle x_{j}^{*}.$

Then

$\displaystyle\|P(x^{*})\|$ $\displaystyle\leq$ $\displaystyle
C\left(\sum_{j=1}^{\infty}|\langle
P_{j}(x^{*}),x_{j}\rangle|^{q}\right)^{\frac{1}{q}}$ $\displaystyle\leq$
$\displaystyle
C\left(\sum_{j=1}^{\infty}\|P_{j}(x^{*})\|^{q}\right)^{\frac{1}{q}}\stackrel{{\scriptstyle\rm
Lemma~{}\ref{l4}}}{{\leq}}3^{\frac{1}{q}}C\|x^{*}\|.$

Thus, the subspace generated by $(x^{*}_{j})_{j=1}^{\infty}$ is complemented
in $Ti^{*}(p,\gamma)$ which implies that $Ti(p,\gamma)$ contains an isomorphic
copy of $\ell_{p}$, a contradiction. ∎

By Lemma 4 and 7 for all $n$ and all normalized block sequences
$(u_{i})_{i=1}^{n}$ in $Ti^{*}(p,\gamma)$ we have
$\|\sum_{i=1}^{n}u_{i}\|\stackrel{{\scriptstyle K}}{{\sim}}n^{1/q}$ for some
$K$. In [JKO] it was shown that spaces with such a property are saturated by
asymptotic-$\ell_{q}$ sequences. An inspection of their proof (of Theorem 3.7)
shows that any block sequence $(x_{i})$ with $\|x_{i}\|_{\infty}\to 0$ is
asymptotic-$\ell_{q}$. Thus the next Proposition follows from the proof of
Theorem 3.7 in [JKO]. We reproduce the proof for completeness, which is
slightly easier in our case.

######  Proposition 9.

Let $1<p<\infty$ and $0<\gamma<3^{-1/q}$. Every normalized block sequence
$(x_{i})_{i=1}^{\infty}$ in $Ti^{*}(p,\gamma)$ satisfying
$\|x_{i}\|_{\infty}\to 0$ is an asymptotic $\ell_{q}$ basic sequence where
$\frac{1}{p}+\frac{1}{q}=1$.

###### Proof.

Let $m\in\mathbb{N},m\geq 2$. Choose $\varepsilon,\delta>0$, and
$\delta^{\prime}$ satisfy

(1)
$0<\varepsilon<\frac{1}{4m3^{1/q}},\quad\delta=\frac{\varepsilon}{6\gamma^{-1}m},\quad
0<\delta^{\prime}<\frac{\delta^{q+1}}{\gamma^{-q}m}.$

Let $M\in\mathbb{N}$ be such that $\|x_{i}\|_{\infty}<\delta^{\prime}$ for all
$i\geq M$. Let $(y_{i})_{i=1}^{m}$ be a normalized block basis of
$(x_{i})_{i\geq M}$. We will show that for all scalars $(a_{i})_{i=1}^{m}$
with $\sum_{i=1}^{m}|a_{i}|^{q}=1$ we have

(2) $\frac{1}{3^{1/q}}\leq\left\|\sum_{i=1}^{m}a_{i}y_{i}\right\|\leq
3^{q+1}\gamma^{-q}.$

Fix $(a_{i})_{i=1}^{m}$. The left hand side inequality holds for all
normalized block vectors and was shown in Lemma 4.

For each $i$, write $a_{i}y_{i}=\sum_{j=1}^{n_{i}+1}y_{i,j}$ where $y_{i,j}$’s
are successive blocks with $\delta\leq\|y_{i,j}\|<\delta+\delta^{\prime}$ and
$\|y_{i,n_{i}+1}\|<\delta$. Then by Lemma 4

$|a_{i}|=\|a_{i}y_{i}\|\geq
3^{-1/q}\Big{(}\sum_{i=1}^{n_{i}+1}\|y_{i,j}\|^{q}\Big{)}^{1/q}\geq
3^{-1/q}\delta n^{1/q}_{i}.$

Thus for all $1\leq i\leq m$,

(3) $n_{i}\leq\frac{3|a_{i}|^{q}}{\delta^{q}}.$

Moreover, by shrinking each $y_{i,j}$ to have norm exactly $\delta$ at a cost
of $\delta^{\prime}$ we have by Lemma 7 that

$\|a_{i}y_{i}\|\leq\gamma^{-1}\delta
n^{1/q}_{i}+n_{i}\delta^{\prime}+\delta\leq\gamma^{-1}\delta
n^{1/q}_{i}+2\delta$

since
$n_{i}\delta^{\prime}+\delta\leq\frac{3}{\delta^{q}}\delta^{\prime}+\delta\leq\frac{3}{\delta^{q}}\frac{\delta^{q+1}}{\gamma^{-q}m}+\delta\leq\frac{\delta}{m}+\delta<2\delta$.

If $|a_{i}|\geq\varepsilon$ then $n_{i}\neq 0$ and from above
$\varepsilon\leq\|a_{i}y_{i}\|\leq\gamma^{-1}\delta n^{1/q}_{i}+2\delta\leq
3\gamma^{-1}\delta n^{1/q}_{i}$ since $\gamma^{-1}n^{1/q}_{i}>1$. Thus

$n^{1/q}_{i}>\frac{\varepsilon\gamma}{3\delta}=2m.$

Let

$N=\sum_{\\{i:|a_{i}|\geq\varepsilon\\}}n_{i}.$

Then $2m\delta<\delta n^{1/q}_{i}\leq\delta N^{1/q}$, and by above
$N\delta^{\prime}<\delta$. We have, using Lemma 7 again,

$\displaystyle\Big{\|}\sum_{\\{i:|a_{i}|\geq\varepsilon\\}}a_{i}y_{i}\Big{\|}$
$\displaystyle\leq\gamma^{-1}\delta N^{1/q}+N\delta^{\prime}+m\delta$
$\displaystyle\leq\gamma^{-1}\delta N^{1/q}+\delta+m\delta$
$\displaystyle\leq\gamma^{-1}\delta N^{1/q}+2m\delta$
$\displaystyle\leq\gamma^{-1}\delta N^{1/q}+\delta N^{1/q}.$

Thus

(4) $\Big{\|}\sum_{\\{i:|a_{i}|\geq\varepsilon\\}}a_{i}y_{i}\Big{\|}\leq
2\gamma^{-1}\delta N^{1/q}.$

On the other hand, by Lemma 4 we have

(5)
$\displaystyle\Big{\|}\sum_{\\{i:|a_{i}|\geq\varepsilon\\}}a_{i}y_{i}\Big{\|}$
$\displaystyle\geq
3^{-1/q}\Big{(}\sum_{\\{i:|a_{i}|\geq\varepsilon\\}}|a_{i}|^{q}\Big{)}^{1/q}\geq
3^{-1/q}(1-\varepsilon m)^{1/q}\geq\frac{1}{2}3^{-1/q},$

and

$\displaystyle\Big{\|}\sum_{\\{i:|a_{i}|<\varepsilon\\}}a_{i}y_{i}\Big{\|}<m\varepsilon\stackrel{{\scriptstyle\ref{ep-
delta}}}{{<}}\frac{1}{4}3^{-1/q}\stackrel{{\scriptstyle\ref{big-coeff-
lower}}}{{\leq}}\frac{1}{2}\Big{\|}\sum_{\\{i:|a_{i}|\geq\varepsilon\\}}a_{i}y_{i}\Big{\|}\stackrel{{\scriptstyle\ref{big-
coeff}}}{{<}}\gamma^{-1}\delta N^{1/q}.$

Thus by the triangle inequality

$\displaystyle\Big{\|}\sum_{i=1}^{m}a_{i}y_{i}\Big{\|}^{q}$
$\displaystyle<3^{q}\gamma^{-q}\delta^{q}N$ $\displaystyle\leq
3^{q}\gamma^{-q}\delta^{q}\sum_{\\{i:|a_{i}|\geq\varepsilon\\}}n_{i}$
$\displaystyle\leq
3^{q+1}\gamma^{-q}\sum_{\\{i:|a_{i}|\geq\varepsilon\\}}|a_{i}|^{q}\quad\text{by}\
(\ref{n_i})$ $\displaystyle\leq 3^{q+1}\gamma^{-q}.$

∎

######  Theorem 10.

Let $1<p<\infty$ and $\gamma>0$ be sufficiently small. Every subsymmetric
basic sequence in the dual space $Ti^{*}(p,\gamma)$ is equivalent to the
subsymmetric canonical basis $(e^{*}_{j})_{j=1}^{\infty}$ which is not
symmetric.

###### Proof.

By Proposition 5 $(e^{*}_{j})_{j=1}^{\infty}$ is not symmetric.

Let $(x^{*}_{j})_{j=1}^{\infty}$ be a normalized subsymmetric basic sequence
in $Ti^{*}(p,\gamma)$. By passing to a subsequence we may assume that
$(x^{*}_{j})_{j=1}^{\infty}$ is a block basis of $(e_{j})_{j=1}^{\infty}$. If
we suppose that $\lim_{j\to\infty}\|x_{j}^{*}\|_{\infty}=0$, then by combining
Lemma 4, Lemma 6 and Proposition 9, we obtain that
$(x^{*}_{j})_{j=1}^{\infty}$ is an asymptotic $\ell_{q}$ basic sequence. Then
the subsymmetry would imply that $(x^{*}_{j})_{j=1}^{\infty}$ is equivalent to
the unit vector basis of $\ell_{q}$ which contradicts Lemma 8.

Thus, by passing again to a subsequence, we may assume that for all
$j\in\mathbb{N}$, $\|x_{j}^{*}\|_{\infty}\geq c$ for some $c>0$. Then
$(x^{*}_{j})_{j=1}^{\infty}$ $c$-dominates $(e^{*}_{j})_{j=1}^{\infty}$. On
the other hand, by Lemma 6 $(x^{*}_{j})_{j=1}^{\infty}$ is 1-dominated by
$(e^{*}_{j})_{j=1}^{\infty}$ and therefore, they are equivalent. ∎

Reflexivity of $Ti(p,\gamma)$ and duality yield the following

######  Corollary 11.

Let $1<p<\infty$ and $\gamma>0$ be sufficiently small. Every subsymmetric
basis of a quotient space of $Ti(p,\gamma)$ is equivalent to the canonical
basis $(e_{j})_{j=1}^{\infty}$.

######  Proposition 12 ([CDKM]).

Let $(e^{*}_{i})$ be a subsymmetric basis which is not equivalent to the unit
vector basis of $\ell_{p}$ or $c_{0}$. Then either $(e_{i})$ or $(e_{i}^{*})$
admits a continuum of non-equivalent subsymmetric block bases.

This, together with Theorem 10, give us the following

######  Corollary 13.

For $1<p<\infty$ and sufficiently small $\gamma$, the basis $(e_{i})$ of
$Ti(p,\gamma)$ has a continuum many non-equivalent subsymmetric block bases.

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11institutetext: Department of Computer Science, The University of Sheffield,
Sheffield, S10 2TN, UK 11email<EMAIL_ADDRESS>
https://www.sheffield.ac.uk

# An incremental hybrid adaptive network-based IDS in Software Defined
Networks to detect stealth attacks

Abdullah H Alqahtani

###### Abstract

Network attacks have became increasingly more sophisticated and stealthy due
to the advances in technologies and the growing sophistication of attackers.
Advanced Persistent Threats (APTs) are a type of attack that implement a wide
range of strategies to evade detection and be under the defence’s radar.
Software Defined Network (SDN) is a network paradigm that implements dynamic
configuration by separating the control plane from the network plane. This
approach improves security aspects by facilitating the employment of network
intrusion detection systems. Implementing Machine Learning (ML) techniques in
Intrusion Detection Systems (IDSs) is widely used to detect such attacks but
has a challenge when the data distribution changes. _Concept drift_ is a term
that describes the change in the relationship between the input data and the
target value (label or class). The model is expected to degrade as certain
forms of change occur. In this paper, the primary form of change will be in
user behaviour (particularly changes in attacker behaviour). It is essential
for a model to adapt itself to deviations in data distribution. SDN can help
in monitoring changes in data distribution. This paper discusses changes in
stealth attacker behaviour. The work described here investigates various
concept drift detection algorithms. An incremental hybrid adaptive Network
Intrusion Detection System (NIDS) is proposed to tackle the issue of concept
drift in SDN. It can detect known and unknown attacks. The model is evaluated
over different datasets showing promising results.

###### Keywords:

SDN APT IDS Machine Learning concept drift.

##  1 Introduction

Software-Defined Networks (SDNs) offer improved network flexibility by
enabling the dynamic configuration of the network infrastructure. This is, in
part, a result of the architectural choice to separate the control and data
planes of the network. The centralisation of the SDN controller provides
visibility over the entire network so monitoring the network and collecting
data from network devices is much easier than in traditional networks [28]. A
number of researchers have sought to use the features of SDN networks to
detect network attacks [1, 2, 21, 48].

However, SDNs bring with them specific threats. For example, SDNs rely on
nodes maintaining flow control rules that tell a node how to handle various
packets. Knowing those rules (or even some of them) can help an attacker. This
gives rise to so-called flow rule reconstruction attacks. Protecting against
such inference attacks remains a challenge. Attackers determined to infer such
information can launch attacks that have characteristics of Advanced
Persistent Threats (APTs), typically seeking to achieve their goals
stealthily, and following a strategy of keeping under any detection thresholds
by employing slow movement and generating or engaging in only a low volume of
traffic. Rather than engage in a ‘smash-and-grab’ as would traditional
attackers, APTs might stay days, weeks, months or years in the victim’s
network and seek to achieve their goals while minimising the risk of raising
suspicion [5, 24].

Nowadays, machine learning (ML) techniques are widely employed by Network
Intrusion Detection Systems (NIDSs) to detect malicious activities inside the
network. In NIDS, known attacks can be detected by _signature-based_ detection
techniques. Such techniques detect predefined patterns of malware or malicious
behaviour. If an attack is unknown, as with most APTs, _anomaly-based_
detection techniques give better chances for detection. In an anomaly-based
detection technique, the model profiles the normal user’s behaviour by
training using only normal traffic (or other behaviour). When an attack
happens, malicious activities are detected as deviations from the pattern
characteristics of profiled normal behaviour.

One of the main challenges in ML-based intrusion detection systems is that the
environment from which data is sampled changes over time. That is the
distribution of the data to be classified changes. Such change may occur, for
example, as a result of a change in the behaviour of both benign users and
attackers. A change in the network environment, e.g. adding more users or
devices to the network, is another reason causing a change in data
distribution. _Concept drift_ is the change over time in the relationship
between the input data and the predicted output (i.e., how inputs are
classified). If the drift is not accounted for, the results of a predictive ML
model will deteriorate [30].

When concept drift occurs, there is a need for the IDS to adapt itself to
change by training over new data; it cannot simply depend on old data. This
paper combines two incremental adaptive techniques to improve the system’s
performance. An Adaptive Random Forests (ARF) (a signature-based detection
technique) and an Adaptive One-Class SVM (an anomaly-based detection
technique) are proposed to work in a hybrid approach to detect known and
unknown attacks respectively. Combining these signature-based and anomaly-
based approaches improves detection and outperforms existing standard ML
techniques. Various experiments are conducted over different datasets to
evaluate the proposed model. The adoption of drift detection and response
strategies counters model degradation due to concept drift. The proposed model
is shown to detect known and unknown stealth attacks in SDN as well as
traditional attacks.

##  2 Background

###  2.1 Concept Drift

The distribution of data an intrusion detector is presented with may change
for a number of reasons, e.g. updating or replacing network devices,
increasing or decreasing the number of users or devices in the network, and
user behaviour changes. A classifier’s performance may degrade if the data
witnessed under training no longer adequately represents the current
situation. The term _concept drift_ is often used to describe this. Concept
drift can be understood as changes in the relationship between the input and
the target outputs of a classifier [30]. It may be sudden/abrupt, incremental,
gradual, or reoccurring (see Figure 1). There are three main categories of
concept drift detection algorithms [30]:

  * •

Error rate-based drift detection: This technique calculates the error rates
raised by the base classifier. A concept drift happens if the error rate is
increased or decreased dramatically. The most popular error rate-based drift
detection algorithms are ADaptive WINdowing (ADWIN) [9], Drift Detection
Method (DDM) [19], Early Drift Detection Method (EDDM) [8], Hoeffding’s bounds
(HDDM_A and HDDM_W) [18], and Kolmogorov-Smirnov Windowing (KSWIN) [40].

  * •

Data distribution-based drift detection: These algorithms use distance
functions to calculate the pattern changes between old and new data. They have
the advantage of reporting the time and location of the drift. However, they
have high computational costs [30] [32]. Examples of these techniques are kdq-
tree [15], CM [32] [31], RD [27], SCD [44], EDE [23], SyncStream [43], PCA-CD
[39], LSDD-CDT [11], LSDD-INC [12], and LDD-DSDA [29].

  * •

Multiple hypothesis test drift detection: These approaches apply multiple
techniques (similar to those in the previous two categories). They apply
different ways to detect the concept drift, but in general, they calculate the
drift by employing different hypothesis tests in a parallel or hierarchical
approach [30].

Figure 1: Concept drift types (adapted from [30])

###  2.2 Incremental learning model

To overcome the issue raised by concept drift (changes in data distribution),
the model should learn incrementally over new instances. In standard machine
learning techniques, the model is typically trained offline, with all training
data available at the time of training the model. The model can be deployed
when the training is completed. It can be trained later using a new batch of
data. This type of training is called batch (or offline) training. The
incremental learning model does not require the entire data to be available
initially. Instead, the data is processed sequentially, and the model updates
itself as new data arrives [20]. Figure 2 illustrates the difference between
the two models.

Figure 2: Difference between batch training and incremental learning model

Our model uses two incremental learning techniques to adapt to concept drift:

  * •

Adaptive Random Forest (ARF): A Random Forest is an ensemble ML technique that
consists of a large number of individual decision trees (DTs). The target
class is the class most voted for by these trees [10]. An Adaptive Random
Forest (ARF) is an optimised technique for Random Forest that enables it to
classify evolving data streams. It employs resampling methods and adaptive
operators to overcome the issues of different types of concept drift. ARF
implements a drift detector that observes the accuracy of the prediction on
every incoming data item. Every tree has a threshold that indicates the change
is a warning or a drift. When a tree is identified as suffering from concept
drift a warning is triggered and a new tree is trained in the background to
replace it [22].

  * •

One-class Support Vector Machine (OC-SVM): An OC-SVM is an unsupervised
learning technique that can be used for anomaly-based detection. It learns to
draw boundaries between normal data and anomalies. In the training phase, the
model is trained solely over normal instances. In the testing (or production)
phase, when a new instance deviates from the normal data, it is considered an
anomaly. OC-SVM is a batch learning model which cannot work on streaming data.
To overcome this issue, Stochastic Gradient Descent (SGD) [41] is augmented to
be used with One-Class SVM as an online version for OC-SVM (giving rise to an
Adaptive One-Class SVM) to detect anomalies in streaming data. SGD computes
the gradient when a single instance (or mini-batch) of data arrives. Another
advantage of SGD is its highly efficient for working on large data [47].

###  2.3 Paper organisation

The rest of this paper is organised as follows. Section 3 summarises the
capabilities of existing proposed approaches. Section 4 shows our
contributions. An explanation of the proposed model is given in section 5.
Evaluation results are presented in section 6. Conclusions and future work are
given in section 7.

##  3 Related Work

Adaptive approaches have been brought to bear for intrusion detection in non-
SDN environments. An ensemble classifier has been proposed to detect concept
drift caused by the existence of new intrusions [51]. _HDDM_ [18] is used as
the concept drift detector and the system scores 0.94 in accuracy. Another
ensemble classifier [33] uses Hoeffding Adaptive Tree and Adaptive Random
Forest to detect network attacks and uses ADWIN for concept drift. A network-
based IDS uses a Page-Hinkley test (PHT) to detect concept drift to improve a
deep learning classifier [6]. The classifier has high accuracy (0.99) and is
applied on the CICIDS2017 [37] dataset (which contains only traditional
attacks). The model is not generalised as it is trained over the dataset
without removing features that could cause over-fitting. A stream learning IDS
[25] is proposed for detecting concept drift. CIDD-ADODNN [38] uses Adadelta
optimiser-based deep neural networks (ADODNN) to classify imbalanced data. The
proposal used anomaly-based detection and was evaluated over three different
datasets. Optimised Adaptive and Sliding Windowing (OASW) [50] is a window-
based drift detection technique proposed for Internet of Things systems. The
authors proposed an optimisation for LightGBM to detect anomalies in such
systems. It is evaluated over IoTID20 dataset [49] and NSL-KDD [36]. The
accuracy results are 0.99 on IoTID20 [49] and 0.98 over the NSL-KDD dataset.
These schemes are not proposed to detect stealth attacks or work on SDN. Table
1 shows a comparison between the aforementioned studies and our proposed
model.

Table 1: Related works Comparison Scheme | SDN | Stealth | Accuracy | Technique |  Dataset
---|---|---|---|---|---
[51] | $\times$ | $\times$ | 0.94 | HDDM |  KDD Cup 99 [26]
[6] | $\times$ | $\times$ | 0.99 | PHT |  CICIDS2017
[25] | $\times$ | $\times$ | 0.96 | bespoke |  Fine-grained Intrusion Dataset (FGD) [42]
[38] | $\times$ | $\times$ | 0.95 | ADWIN |  NSL-KDD
|  |  | 0.93 |  |  Spam [45]
|  |  | 0.76 |  |  Chess[14]
[33] | $\times$ | $\times$ | - | ADWIN |  KDD Cup 99
[50] | $\times$ | $\times$ | 0.99 | ADWIN |  IoTID20
|  |  | 0.98 |  |  NSL-KDD
Proposed model | $\checkmark$ | $\checkmark$ | 0.99 | ADWIN |  APT-SDNdataset [7]
|  |  | 0.98 |  |  DAPT 2020 [35]
|  |  | 0.99 |  |  CICIDS 2017 [37]
|  |  | 0.94 |  |  InSDN [16]

##  4 Contributions

The review of section 3 reveals a lack of work dealing with concept drift in
SDN, especially when attacks are stealthy. Our hybrid model in [3] can detect
such attacks in a non-streaming environment. In this paper, we propose to
detect such attacks in a streaming environment.

Our focus on adaptivity is driven in part by the fact that APTs may engineer
concept drift as a stealth strategy. They may continually adapt their
behaviour so that they maintain sufficient similarities with normal behaviours
to remain undetected [13]. They may stay dormant for a period of time or
employ sleep functions between their activities [3]. Moreover, the majority of
these attacks go through multiple stages and at every stage, they have one or
more tools that may create new classes of attack. New patterns may emerge as a
result of these changes [24] [5] [34]. It is essential to retrain the ML-based
NIDS to track the concept drift and make the model more adaptive to any
changes in data distribution. In our experiments, we use various datasets to
evaluate the proposed model (when attacker changes their behaviour).

The contributions of this paper are:

  * •

The first proposal of incremental adaptive hybrid IDS to overcome concept
drift caused by stealth attackers in SDN.

  * •

A comparison between the proposed hybrid model (adopting concept drift
detection) and standard hybrid model without implementing concept drift
detection techniques.

  * •

Evaluation of the most popular concept drift detection techniques.

  * •

Evaluation of the proposed system over different datasets, reflecting various
attacks and scenarios (including known and unknown attacks).

  * •

Determination of optimal parameters using hyper-parameter tuning.

##  5 Proposed Scheme

Given a period of time [0,$t$], streaming data arrives as a sequence of
packets $K_{0,t}=\\{s_{0},s_{1},s_{2},...,s_{n}\\}$, where
$s_{i}=(X_{i},y_{i})$ is one sample. $X_{i}$ is a vector of independent
features which can be labelled appropriately with $y_{i}\in\\{0,1\\}$. Let
$P_{t}(X,y)$ represent the _joint distribution_ of $X$ and $y$ at time $t$.
(It is a probability density function.) Concept drift happens when
$P_{t}(X,y)$ changes over time. Thus, it occurs between two time points
$t_{0}$ and $t_{1}$ if $\exists X\cdot P_{t0}(X,y)\neq P_{t1}(X,y)$ [20]. This
is defined in terms of the _data_ sampled. For example, if a new attack is
developed after time $t_{0}$ and launched, corresponding to $X^{\prime}$ say,
then $P_{t0}(X^{\prime},1)=0$ (the attack never occurs in the data and so has
zero density) but $P_{t1}(X^{\prime},1)\neq 0$ (the attack does occur and so
has non-zero density). Technically, a specific vector of features $X$ could in
some circumstances correspond to malicious action but be benign in others,
though often the assignation will be unequivocal. ML-based models that seek to
predict $y$ from $X$ must adapt to ensure continued high performance.

The proposed hybrid system combines signature-based and anomaly-based
detection techniques using Adaptive Random Forest (ARF) and adaptive One-Class
SVM. The model monitors networks through the SDN controller. Figure 3 shows
the place of the proposed NIDS. The details of the detection of the model is
illustrated in Figure 4. Every instance that arrives at the system is,
firstly, examined by the signature-based ARF module (discussed in section
5.1). If the module classifies that instance as a known attack, then there is
no need for further investigation by the anomaly-based detection module. If
the packet is classified as benign by the signature-based module, it is passed
to the anomaly-based detection module (Adaptive One-class SVM) (discussed in
section 5.2) to check if it is an unknown attack. The administrator is
notified if the signature or anomaly-based module flags an instance as an
attack. The concept drift is detected using ADWIN, a concept drift detector,
in both classifiers, and the model is updated. The model can detect known and
unknown attacks. Both are incremental and can adapt themselves against concept
drifts.

Figure 3: The proposed NIDS model architecture Figure 4: The proposed adaptive
hybrid model

###  5.1 Signature-based detection module

The first detection phase is the signature-based detection module, where the
classifier is trained over labelled data to learn the pattern of malicious and
normal activities. Adaptive Random Forest (ARF) is employed as the incremental
adaptive technique. The ARF uses a drift detector in every tree, which
monitors drifts and warnings. In the case of a warning, the trees in the
background will train new trees. If a drift is detected, the primary model
will be replaced by the trained trees. Hyper-parameter tuning is applied to
ensure the model is generalised (avoiding overfitting or underfitting). Table
2 shows the selected parameters of ARF.

###  5.2 Anomaly-based detection module

Adaptive One-class SVM is an anomaly detection algorithm. The classifier is
trained over the normal data and tested over both normal and malicious
samples. During the training, it develops a profile of normal traffic. If the
incoming traffic differs from the normal data that it is trained on, it is
considered an attack. It also adapts itself incrementally when the data
distribution is changed. The new data points are compared to the existing
model. A change in the distribution of the data points indicates a concept
drift is happening. SVMs and their variants have been shown to be highly
effective classifiers across many domains. The selected parameters after
applying hyper-parameter tuning are shown in Table 2.

Table 2: Selected parameters of ARF and OC-SVM parameter |  description | optimal value
---|---|---
AdaptiveRandomForestClassifier (ARF)
max_features |  Maximum number of features in every single run | 5
n_estimators |  Number of trees | 7
drift_detection_method |  concept drift detection technique | ADWIN
Adaptive One-Class SVM (OC-SVM)
$\nu$ |  This controls the fraction of outliers in the system | 0.2
Gamma |  Kernel coefficient | 0.9

##  6 Evaluation and Results

APT campaigns go through multiple stages and different types of attacks are
involved in every stage. Moreover, there is no one pattern for APT attacks
[5]. In this paper, a number of experiments were conducted on different
datasets to show the effectiveness of the proposed model in detecting attacks
and dealing with concept drift (when attacker changes their behaviour). These
datasets include different attacks and scenarios that present a change in
attacker behaviour. Table 3 presents a summary of every dataset and the aim of
conducting the experiment over that dataset. In all experiments, various
evaluation standard measures are recorded: _Accuracy, Recall, Precision, and
F1-score_.

Table 3: A summary of datasets and experiments objectives Dataset |  Dataset feature |  aims of the experiment
---|---|---
DAPT 2020 [35] |  Contains APT traffic and a change in attacker behaviour (concept drift) |  \- evaluate the model in detecting APTs
|  |  \- evaluate the model against concept drift
|  |  \- compare the standard ML techniques with those after adaptation
APT-SDNdatase [7] |  stealth attacks on SDN (reconstructing flow rules in SDN) |  evaluate the model on detecting stealth attacks in SDN
InSDN [16] |  Traditional attacks in SDN over different days |  \- evaluate the model against various attacks in SDN
|  |  \- evaluate the model against concept drift over SDN network
CICIS 2017 [37] |  A benchmark IDS dataset has various attacks over different days |  \- evaluate the model against different traditional attacks
|  |  \- evaluate the model against concept drift

  1. 1.

Model evaluation over APT dataset: DAPT 2020 [35] is an APT-based dataset used
in this experiment to evaluate the proposed model against APT attacks and
attackers’ behaviour changes (concept drift). The approach of conducting
normal traffic and various attacks over different days makes this dataset very
useful for evaluation. In Appendix A, Table 6 presents details of the attack
and normal traffic scenario in the dataset. The dataset is prepared by
removing columns that can affect the system’s performance. Flow ID, Source IP,
Source Port, Destination IP, Destination Port and Timestamp are the features
that are removed from the dataset. Null values are replaced by zeros, and
labels are converted to (0) and (1) as the system is a binary classifier. The
dataset’s features are scaled using scikit-learn’s StandardScaler [46]. The
first experiment compares error rate-based drift detection techniques: ADWIN,
DDM, EDDM, HDDM_A, HDDM_W, PageHinkle, and KSWIN. The system gives significant
results for detecting APTs, scoring 0.98 in Accuracy, Recall, Precision and F1
using any of these techniques (concept drift detector). Table 4 shows the
results of all techniques with a negligible variation in scores.

Table 4: Results for error rate-based drift detection (over DAPT 2020 dataset) Tech | Accuracy | Recall | Precision | F1
---|---|---|---|---
ADWIN | 0.9871 | 0.9882 | 0.9856 | 0.9869
DDM | 0.9872 | 0.98871 | 0.9853 | 0.9870
EDDM | 0.9862 | 0.9879 | 0.9839 | 0.9859
HDDM_A | 0.9867 | 0.9888 | 0.9842 | 0.9865
HDDM_W | 0.9838 | 0.9852 | 0.9818 | 0.9835
PageHinkle | 0.9855 | 0.9879 | 0.9826 | 0.9852
KSWIN | 0.9856 | 0.9872 | 0.9834 | 0.9853

Another experiment is conducted to evaluate the implementation of the
distribution-based detection technique (using kdq-tree). This achieves 0.83,
0.80, 0.84 and 0.82 in Accuracy, Recall, Precision and F1 respectively. The
results are not significant, but the technique has advantages as it is
unsupervised.

ADWIN will be used on the model as the concept drift detection technique.
Because most error-based concept drift techniques are very similar but ADWIN
has more flexibility than others. The window size (in ADWIN) is variable as it
can keep expanding and shrinking, making the sliding window mechanism more
flexible than other techniques [9] [30]. In addition, it gives better results
than the distribution-based concept drift algorithm (e.i. kdq-tree).

To evaluate to which extent adopting concept drift detection can improve the
system, a further experiment is conducted on standard classifiers (batch
learning). The standard classifiers Random Forest and One-Class SVM are
employed to form the hybrid system. The results are 0.72 Accuracy, 0.72
Recall, 0.78 Precision and 0.75 in F1. Figure 5 shows how the system is
improved after implementing the adaptivity (using ADWIN for error rate-based
drift detection and kdq-tree for data distribution based).

Figure 5: A comparison on the evaluation of batch learning and the proposed
incremental model over the DAPT 2020 dataset

  2. 2.

Model evaluation over stealth SDN-based dataset: These experiments aim to
evaluate how well the proposed hybrid model detects stealth attacks over SDN
network. APT-SDNdataset [7] is the dataset used in the experiments. The
dataset has a probe scan that reconstructs flow rules in a stealthy manner
[4]. The first experiment uses ADWIN as the concept drift detector technique.
It achieves with 0.99 in accuracy and recall. 0.95 and 0.97 in precision and
F1 respectively.

The second experiment uses the data distribution-based drift detection
technique. The kdq-tree scores 0.99, 0.96, 0.96 and 0.96 in accuracy, recall,
precision and F1 respectively. The results show that the error-based detection
algorithm (ADWIN) performs better than kdq-tree. Precision (which affects F1)
is much lower in kdq-tree, compared to ADWIN results, due to the high
percentage of false positives.

  3. 3.

Model evaluation over SDN-based dataset: This experiment is conducted over an
SDN-based dataset to show how the model can detect various attacks over SDN
networks and adapt the model over the concept drift issue. InSDN [16] is a
dataset made up from SDN-based traffic. It is divided into three groups based
on the type of traffic and the targeted machine. The _normal group_ : contains
just normal traffic with no attacks; the _Metasploitable-2 group_ : includes
attacks that targeted the Metasploitable-2 machine; and the _OVS group_ :
contains attacks that target the OVS switch. The attacks in the dataset are
DoS, DDoS, web attacks(XSS and SQL injection), R2L, Malware(botnet), probe and
U2R (exploitation) attacks. In Appendix B, Table 7 gives a summary of attack
details and the normal user behaviour. More details about the dataset are in
[16]. Firstly, the model is evaluated using an error-based concept drift
detection technique. ADWIN is used in the experiment scoring 0.94 in accuracy,
1 in recall, 0.93 in precision and 0.96 in F1. Another experiment is conducted
using the distribution-based technique. kdq-tree is the algorithm used in the
experiment, and the results are 0.94, 0.99, 0.93 and 0.96 in accuracy, recall,
precision and F1, respectively. The results show significant performance in
detecting actual attacks correctly (True Positives). But this came with a
price, as it classified some normal instances as attacks (False Positives).

  4. 4.

Model evaluation over traditional attacks: A benchmark and recent dataset in
the intrusion detection systems is used in this experiment to evaluate the
model against traditional attacks and concept drift problems. We use the
CICIDS 2017 dataset in the experiment. It has a large amount of data recorded
over five days and contains different types of attacks. In Appendix C, Table 8
shows the normal and attack details. The first day is just normal traffic, but
the following days have different attacks every day. This is a suitable
scenario for concept drift as the attacker changes their behaviour over time.
In this work, the dataset is prepared by removing all features that may expose
the identity of the sender/receiver or cause over-fitting to the system. The
removed features are Flow ID, Source IP, Source Port, Destination IP,
Destination Port and Timestamp. The feature importance technique is applied
using Random Forest [17]. Just seven features, the most important of which are
shown in Figure 6, were selected and applied. The proposed system is a binary
classifier. Thus all attacks are converted to one class (1), and all normal
records are assigned to (0). In the first experiment, ADWIN is used as the
concept drift detector. The second experiment uses the concept drift
distribution-based detector (kdq-tree). Significant results are shown in both
experiments. All metrics score 0.99 while implementing ADWIN. With kdq-tree,
it has very high scores in accuracy (0.99) and recall (1); however, the result
in precision is 0.86 and in F1 is 0.92.

Figure 6: CICIDS 2017 feature importance

Table 5 presents a summary of the results of all experiments applying ADWIN
and kdq-tree on different datasets.

Table 5: Proposed model results over different datasets Technique | Dataset | Accuracy | Recall | Precision | F1
---|---|---|---|---|---
ADWIN | APT-SDNdataset | 0.99 | 0.99 | 0.95 | 0.97
kdq-Tree | APT-SDNdataset | 0.99 | 0.96 | 0.96 | 0.96
ADWIN | InSDN | 0.94 | 1 | 0.93 | 0.96
kdq-Tree | InSDN | 0.94 | 0.99 | 0.93 | 0.96
ADWIN | CICIDS 2017 | 0.99 | 0.99 | 0.99 | 0.99
kdq-Tree | CICIDS 2017 | 0.99 | 1 | 0.86 | 0.92
ADWIN | DAPT 2020 | 0.98 | 0.98 | 0.98 | 0.98
kdq-Tree | DAPT 2020 | 0.83 | 0.80 | 0.84 | 0.82

##  7 Conclusion and future works

Machine learning model performance will typically degrade when the data
distribution is changed over time. APTs usually change their behaviour,
causing a change in data distribution. An incremental adaptive hybrid
intrusion detection system is proposed in this work. The signature-based
detection (implemented by Adaptive Random Forest) and anomaly-based detection
model (implemented using Adaptive One-Class SVM) are combined to detect
anomalies. It can detect known and unknown stealth attacks. The system adapts
itself incrementally to changes in data distribution (concept drift). ADWIN is
adopted in the proposed model to detect concept drift and adapt the detection
models.

The diversity of datasets used in the evaluation and the high scores obtained
show how the proposed model can detect known and unseen attacks and adapt
against concept drift. We recommend adapting security solutions (e.g.
intrusion detection systems) incrementally. This can maintain the performance
of the system even when a new attack has emerged or the attacker changes their
strategy in the attack. SDN helps in this approach by making gathering and
monitoring data easier.

Our proposed model can deal with concept drift that may occur based on common
system evolution. Examples of these changes are producing new services,
installing or removing a number of network devices or a change in the number
of users. However, our proposed model is not evaluated over this scenario. In
the future, we plan to evaluate the proposed model over these scenarios.
Another plan is to evaluate the system using a fuller range of the most
popular distribution-based concept drift techniques (e.g. CM [32] [31], RD
[27], SCD [44], EDE [23], SyncStream [43], PCA-CD [39], LSDD-CDT [11], LSDD-
INC [12], and LDD-DSDA [29]).

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## Appendix A

Details of the scenario of attacks and normal traffic in DAPT 2020 dataset are
shown in Table 6.

Table 6: DAPT 2020 dataset attacks scenario day | Benign | attack stage |  attack type | attack time
---|---|---|---|---
1 | $\checkmark$ | $\times$ |  - | -
2 | ✓ | Reconnaissance |  scanning | 8:00 a.m. - 6:00 p.m.
3 | $\checkmark$ | Foothold establishment |  Sqli, XSS, authentication bypass | 8:00 a.m. - 6:00 p.m.
4 | $\checkmark$ | Lateral Movement |  network scan (insider), authentication bypass, SQLi | 8:00 a.m. - 6:00 p.m.
5 | $\checkmark$ | Data exfiltration |  Exfiltrating data to C&C | 8:00 a.m. - 6:00 p.m.

## Appendix B

Table 7 shows details of the attack scenario and normal traffic in InSDN
dataset.

Table 7: Attacks details on InSDN dataset day | Data group | attack type | attack time
---|---|---|---
1 | OVS | Web attack | 25/12/2019 (17:00 - 23:00)
2 | OVS | Probe | 26/12/2019 ( 13:00 - 15:00)
3 | OVS | DoS | 27/12/2019 (16:00 - 19:00)
4 | OVS | Botnet | 31/12/2019 (14:00 - 15:00)
| OVS | DDoS | 31/12/2019 (18:00 - 19:00)
| Normal | - | 31/12/2019 (14:00 - 15:00)
5 | OVS | DoS | 01/01/2020 (22:00 - 24:00)
6 | Metasploitable-2 | DoS | 09/01/2020 (19:00 - 20:00)
| Metasploitable-2 | Probe | 09/01/2020 (16:00 - 18:00)
| Metasploitable-2 | U2R | 09/01/2020 (21:00 - 22:00)
7 | Metasploitable-2 | BFA | 10/01/2020 (01:00 - 04:00)
| Metasploitable-2 | DDoS | 10/01/2020 (05:00 - 06:00)
| Metasploitable-2 | U2R | 10/01/2020 (00:00 - 21:00)
8 | OVS | BFA | 12/01/2020 (01:00 - 02:00)
| OVS | DDoS | 12/01/2020 (16:00 - 16:00)
9 | Normal | - | 04/02/2020 ( 12:00 - 13:00)
| Normal | - | 04/02/2020 ( 12:00 - 13:00)
10 | Normal | - | 05/02/2020 ( 11:00 - 20:00)
11 | Normal | - | 07/02/2020 ( 13:00 - 20:00)

## Appendix C

In this section, Table 8 shows the normal and attack details of CICIDS 2017
dataset.

Table 8: Attacks details on CICIDS 2017 dataset day | Benign | attack category | attack type | attack time
---|---|---|---|---
1 | $\checkmark$ | $\times$ | - | -
2 | $\checkmark$ | Brute Force | FTP-Patator | 9:20 – 10:20 a.m.
|  |  | SSH-Patator | 14:00 – 15:00 p.m.
3 | $\checkmark$ | DoS / DDoS | DoS slowloris | 9:47 – 10:10 a.m.
|  |  | DoS Slowhttptest | 10:14 – 10:35 a.m.
|  |  | DoS Hulk | 10:43 – 11 a.m.
|  |  | DoS GoldenEye | 11:10 – 11:23 a.m.
|  | Heartbleed | Heartbleed Port 444 | 15:12 - 15:32
4 | $\checkmark$ | Web Attack | Brute Force | 9:20 – 10 a.m.
|  |  | XSS | 10:15 – 10:35 a.m.
|  |  | Sql Injection | 10:40 – 10:42 a.m.
|  | Infiltration – Dropbox download | Meta exploit Win Vista | 14:19 & 14:20-14:21 p.m.
|  |  |  | 14:33 -14:35
|  | Infiltration – Cool disk | MAC | 14:53 p.m. – 15:00 p.m
|  | Infiltration – Dropbox download | Win Vista | 15:04 – 15:45 p.m.
5 | $\checkmark$ | Botnet ARES |  | 10:02 a.m. – 11:02 a.m.
|  | probe | port scan | 13:55 - 14:35
|  |  |  | 14:51 - 15:29
|  | DDoS | LOIT | 15:56 – 16:16