Detecting misbehavior in distributed wireless interference networks

Abstract

The paper analyzes the case of K distributed interference coupled point-to-point links, which utilize a general power update algorithm, which converges monotonically. The channels are constant for the duration of the algorithm. In such a system we allow a set of links to have the ability to misrepresent their utilities, i.e. the receiver sends a misrepresented utility to its transmitter. It is shown that the system, i.e., the set of correctly behaving links, has the ability to detect such a behavior, if and only if G _Restricted is irreducible. Here G _Restricted is the restricted global dependency matrix, which captures the effect of interference coupling in the system. We then analyze the special case of the Foschini-Miljanic power update algorithm. After at most K − 1 steps, all links are able to detect the misbehavior. Example interference networks are discussed to illustrate the results. Finally the applicability of the results to the practically relevant cases of concave, convex and log-convex interference functions is displayed.

Appendices

Appendix 1: Interference function framework

Let $\mathcal{P}$ be the set of all power vectors. In our paper, we have ${\mathcal{P}:= \mathbb{R}_{+}^{K+1}}$ unless explicitly mentioned otherwise.

Definition 7

We say that ${\mathcal{I}: \mathcal{P}\mapsto\mathbb{R}_{+}}$ is an interference function, if the following axioms are fulfilled:

$$ \begin{aligned} \begin{array}{ll} A1 & \hbox{conditional positivity}\,{\mathcal{I}}({\user2{\underline{p}}}) > 0\, {\rm if} \,{\user2{\underline{p}}} > {\bf 0}\\ A2 & \rm{scale\ invariance}\,{\mathcal{I}}(\alpha{\user2{\underline{p}}}) = \alpha{\mathcal{I}}({\user2{\underline{p}}})\,\rm{for\,all}\, \alpha > 0\\ A3 & \rm{monotonicity}\,{\mathcal{I}}({\user2{\underline{p}}}) \geq {\mathcal{I}}({\hat{\user2{\underline{p}}}})\,\rm{if}\, {\user2{\underline{p}}} \geq {\hat{\user2{\underline{p}}}}\\ A4 & \hbox{strict monotonicity}\,{\mathcal{I}}( \user2{\underline{p}}) > {\mathcal{I}}(\hat{\user2{\underline{p}}})\,\rm{if}\,{\underline{\bf p}} \geq {\hat{\user2{\underline{p}}}},\\ & \quad p_{K+1} > \hat{p}_{K+1}. \end{array} \end{aligned} $$

Note that we require that $\mathcal{I}( \user2{\underline{p}})$ is strict monotone with respect to the last component p _K+1. An example is $\mathcal{I}(\user2{\underline{p}}) = {\bf v}^{T}\user2{p} + \sigma^{2}$ where ${{\bf v}\in \mathbb{R}_{+}^{K}}$ is a vector of interference coupling coefficients. The axiomatic framework A1-A4 is connected with the framework of standard interference functions [11]. The details about the relationship between the model A1 - A4 and the Yates’ standard interference functions were discussed in [18] and further investigated in [14]. For the purpose of this paper it is sufficient to be aware that there exists a connection between these two models and the results of this paper are applicable to standard interference functions.

1.1 Concave interference functions

Theorem 3

For each concave interference function $\mathcal{I}$ there exists exactly one convex closed set ${\mathcal{V}_{\mathcal{I}}\subseteq \mathbb{R}_{+}^{K}}$, such that

$$ {\mathcal{I}}(\user2{p}) = \min_{{\bf v}\in {\mathcal{V}}_{{\mathcal{I}}}}\sum_{k=1}^{K}v_{k}p_{k}. $$

(11)

Remark 6

The set ${\mathcal{V}_{\mathcal{I}}}$ can be explicitly constructed [21].

In (11) each concave interference function can be interpreted as an optimization of a receiver filter. We shall choose the optimal receiver filter, such that the corresponding SINR is maximized. ${\mathcal{V}_{\mathcal{I}}}$ is the set that is the domain of linear filter coefficients. This is a generalization of beamforming.

Remark 7

Consider the case of receivers, which can be written as a parametrization of linear receiver filters and maximizes the SINR. All such receivers results in a concave interference function and vice versa.

For each $\user2{p}\gg {\bf 0}$ the set ${\mathcal{V}_{\mathcal{I}}}$ (${{\hat{\bf v}} = {\hat{\bf v}}(\user2{p})\in\mathcal{V}_{\mathcal{I}}}$) with $\mathcal{I}(\user2{p}) =\sum_{k=1}^{K}\hat{v}_{k}(\user2{p})p_{k}$ is not empty. We shall denote this set with $\mathcal{ORS}(\user2{p}):=$ the set of optimal receiver strategies.

The only assumption we make with respect to this set is that it is rich enough with respect to receiver strategies. For all ${\hat{\user2{p}}} > {\bf 0}$ and for all $k\in [1, \ldots, K]$ and for all δ > 0, we should have $\mathcal{ORS}({\hat{\user2{p}}}) \cap\mathcal{ORS}({\hat{\user2{p}}} + \delta{\bf e}_{k}) = \emptyset$. Therefore, for an receive filter an optimal power allocation ${\hat{\user2{p}}}$ exists, such that ${\hat{\user2{p}}} +\delta{\bf e}_{k}, \delta > 0, 1\leq k\leq K$ is not optimal.

Definition 8

A receiver is said to be fully adaptive, if for all ${\hat{\user2{p}}} > {\bf 0}$ and for all $k\in \mathcal{K}$ and for all δ > 0, we should have $\mathcal{ORS}({\hat{\user2{p}}}) \cap\mathcal{ORS}({\hat{\user2{p}}} + \delta{\bf e}_{k}) = \emptyset$.

Example 5

Optimal beamforming is an example of a fully adaptive receiver strategy.

Theorem 4

Let the receiver be fully adaptive (according to Definition 8). Then, the corresponding interference function is strictly monotonically increasing.

Proof

We shall prove the theorem indirectly. Let the receiver be fully adaptive and the corresponding interference function is not strictly monotonically increasing. Then, there exists a ${\hat{\user2{p}}} > {\bf 0}, \hat{k}\in [1, \ldots, K]$ and $\hat{\delta} > 0$ such that $\mathcal{I}({\hat{\user2{p}}} +\hat{\delta}{\bf e}_{\hat{k}}) = \mathcal{I}({\hat{\user2{p}}})$ for $0\leq \delta\leq \hat{\delta}$. We now choose ${\hat{\bf v}}\in\mathcal{ORS}({\hat{\user2{p}}})$ arbitrarily and $\delta\in [0,\hat{\delta}]$.

$$ \begin{aligned} {\mathcal{I}}({\hat{\user2{p}}}) = & \min_{{\bf v}\in {\mathcal{V}}_{{\mathcal{I}}}}\sum_{k=1}^{K}v_{k}\hat{p}_{k} = \sum_{k=1}^{K}\hat{v}_{k}\hat{p}_{k}\\ = & {\mathcal{I}}({\hat{\user2{p}}} + \delta{\bf e}_{\hat{k}})\\ = & \min_{{\bf v}\in {\mathcal{V}}_{{\mathcal{I}}}}\big(\sum_{k=1}^{K}v_{k}\hat{p}_{k} + v_{\hat{k}}\delta\big) \end{aligned} $$

(12)

Therefore, ${{\hat{\bf v}}\in \mathcal{V}_{\mathcal{I}}}$ and ${\hat{\bf v}}\in \mathcal{ORS}({\hat{\user2{p}}} +\delta{\bf e}_{\hat{k}})$. Therefore, we have that the receiver is not fully adapted. We have our required contradiction.

1.2 Convex and log-convex interference functions

When $\mathcal{I}$ is an interference function, which is convex with respect to p, then $\mathcal{I}(e^{{\bf s}})$ is log-convex with respect to s. When $\mathcal{I}(\user2{p})$ is strictly monotonic on the dependency set L with respect to p, then $\mathcal{I}(e^{{\bf s}})$ is strictly monotonic with respect to s and vice versa. Therefore, it is sufficient to consider the case of log-convex interference functions (the convex case is then only a special case). From the monotonicity property of interference functions it holds for all p > 0, for all μ > 0 and for all k that

$$ \begin{aligned} {\mathcal{I}}_{k}((1 +\mu)\user2{p}) &= {\mathcal{I}}_{k}([p_{k}, (1+\mu)\user2{p}_{-k}])\\ & = (1 + \mu){\mathcal{I}}_{k}([\frac{p_{k}}{1 + \mu}, \user2{p}_{-k}])\\ & \leq (1 + \mu ){\mathcal{I}}_{k}([p_{k}, \user2{p}_{-k}])\\ & = (1 + \mu ){\mathcal{I}}_{k}(\user2{p}) \end{aligned} $$

Hence, we have sub-linear growth with respect to the user k for the scaling of its corresponding power vector input to its interference function.

Definition 9

$\mathcal{I}_{k}$ is local worst case scaling for power vector p with respect to link $k\in \mathcal{K}$, if there exists a μ > 0, such that $\mathcal{I}_{k}([p_{k}, (1 +\mu)\user2{p}_{-k}]) = (1 + \mu)\mathcal{I}(\user2{p})$.

Remark 8

From the perspective of link k if interference function $\mathcal{I}_{k}$ (i.e. the interference experienced by user k) satisfies the property of worst case scaling, then as the name suggests it is the worst possible scaling of the interference experienced by link k, i.e. from the perspective of the γ _k (since $\gamma_{k}(\user2{p}) =p_{k}/\mathcal{I}_{k}(\user2{p})$, where γ _k is the SINR of user k, which links usually would like to maximize. Hence, it is a not necessarily a positive property to be satisfied for a particular link k.

When $\mathcal{I}_{k}$ is convex and local worst case scaling with respect to p, then we have $\mathcal{I}_{k}([x,\user2{p}_{-k}]) = \mathcal{I}_{k}(\user2{p})$ for 0 ≤ x ≤ p _k . Then, $g_{k}(x) = \mathcal{I}_{k}([x, \user2{p}_{-k}])$ is monotonically increasing and convex in x. We have for $0\leq\mu\leq \hat{\mu}$, $\hat{\mu} > 0$, that

$$ (1+ \mu ){\mathcal{I}}_{k}\left(\left[\frac{p_{k}}{1+ \mu}, \user2{p}_{-k}\right]\right) = (1+ \mu ){\mathcal{I}}_{k}(\user2{p}). $$

Therefore, for $\frac{p_{k}}{1+ \hat{\mu}} \leq x\leq p_{k}$ the function g _k with respect to x is constant. Since, g _k is monotonically increasing, it must hold for all x ≤ p _k .

Theorem 5

Let $\mathcal{I}$ be a convex interference function such that property of worst case scaling is not satisfied for each link $k\in \mathcal{K}$ and for all feasible power vectors. Then, $\mathcal{I}$ is strictly monotonically increasing.

Proof

Let $\mathcal{I}$ not satisfy the property of strict monotonicity. Therefore, there exists a $\hat{p} > 0, \hat{\delta}> 0, k\in [1, \ldots, K]$ such that $g_{\hat{k}}(\delta) =\mathcal{I}([{\hat{\user2{p}}}, \delta])$ is constant for $0\leq\delta\leq \hat{\delta}$. We only set $\tilde{\user2{p}} =[{\hat{\user2{p}}}, \hat{\delta}]$. Then, $f_{\hat{k}} =\mathcal{I}([{\tilde{\user2{p}}}_{-k}, x])$ is constant for $0\leq x\leq\hat{p}_{k} + \hat{\delta}_{k} = \tilde{p}_{k}$. Then $\mathcal{I}$ fulfills the worst case scaling property for the power vector ${\tilde{\user2{p}}}$. We have our required contradiction and the result is proved.

$\mathcal{I}$ is an interference function and L is the corresponding dependency set for a particular link. Therefore, for all $k\in L$, there exists a strict positive vector ${\hat{\user2{p}}} ={\hat{\user2{p}}}(k)$ such that the function

$$ {\mathcal{I}}([{\hat{\user2{p}}}, \delta ])=: f_{k}([\delta, {\hat{\user2{p}}}]) $$

(13)

is not constant for all δ > 0. We can without loss of generality, assume that $L = [1, \ldots, K]$. In comparison to Definition 5, we can now say that $\mathcal{I}$ is called strictly monotonic on a dependency set, if for all p » 0 and for all $k\in L$ the function in (13) with respect to δ is strictly monotonically increasing.

Appendix 2: power graph

Let $\mathbf{P}$ and $\mathbf{I}$ represent the set power nodes and the set of interference nodes, respectively. In the definition below, a power node $p_{k}\in \mathbf{P}$ could be thought of as an abstraction of the transmit power of user k. Similarly, an interference node $\mathcal{I}_{k}\in \mathbf{I}$ could be thought of as an abstraction of the interference experienced by user k. Power nodes are not connected to each other. Interference nodes are not connected to each other. If node p _k has an undirected edge to certain other interference nodes, e.g. $\mathcal{I}_{k_{1}},\mathcal{I}_{k_{2}},$ where $k, k_{1}, k_{2}\in \mathcal{K},$ then it implies that user k has the ability to cause interference to users k ₁ and k ₂. Having provided this basic intuition we are now in a position to introduce the power and interference bipartite graphs.

Definition 10

For any given matrix ${\user2{G}\in\mathbb{R}_{+}^{K\times K}, }$ let $\tilde{\mathcal{G}}(\user2{G})$ be the (undirected) bipartite graph of 2K nodes divided into two disjoint sets $\mathbf{P}$ and $\mathbf{I}$ (each of cardinality K) such that, there is no edge between nodes within each of these groups and there is an undirected edge between $p_{l}\in \mathbf{P}$ and $\mathcal{I}_{k}\in \mathbf{I}$ if and only if [G] _kl  > 0.

Now, we introduce the dependency set L _k ′, where L _k ′ = {k}∪ L _k .

Definition 11

An elementary power step is a connection between any two power nodes in the bipartite graph through an interference node.

We can construct an elementary power step between two power nodes p _t and p _s if and only if $t, s \in L_{k}^{'},$ where L _k ′ = {k}∪ L _k (see Fig. 3).

Fig. 3

Elementary power step in a Power graph; p _r and p _s are in L ^′ _k

Definition 12

We call a bipartite graph of 2K nodes divided into two disjoint sets $\mathbf{P}$ and $\mathbf{I}$ (each of cardinality K) a power graph if it consists only of elementary power steps. A power path is a path in the power graph made up one or more elementary power steps.

If the power graph is fully connected then each power node can be reached by every other power node. An example of a fully connected power graph is displayed in Fig. 2. An example of a power graph, which is not fully connected is displayed in Fig. 4. For further details on the topic, kindly refer to [19].

Fig. 4

Example of a power graph, which is not fully connected; L ^′ _k  = {k} ∪ L _k