Edge-Cut Bounds on Network Coding Rates

Active networks are network architectures with processors that are capable of executing code carried by the packets passing through them. A critical network management concern is the optimization of such networks and tight bounds on their performance serve as useful design benchmarks. A new bound on communication rates is developed that applies to network coding, which is a promising active network application that has processors transmit packets that are general functions, for example a bit-wise XOR, of selected received packets. The bound generalizes an edge-cut bound on routing rates by progressively removing edges from the network graph and checking whether certain strengthened d-separation conditions are satisfied. The bound improves on the cut-set bound and its efficacy is demonstrated by showing that routing is rate-optimal for some commonly cited examples in the networking literature.

APPENDIX VALIDITY OF THE PDE BOUND

We prove the validity of the PdE bound described in Sections 5 and 6 for noisy as well as noise-free networks. This section assumes familiarity with advanced concepts in information theory. Recall that we consider the following objects.

• ɛ _d : a set of edges

• S _d: a set of source indices

• π(.): a one-to-one mapping from $\{1,\,2, \ldots,|\cal{S}_d |\}$ to S _d.

• a nonempty subset of $\{\hat W_k^{(i)} :i=1,2, \ldots,D_k\}$ for all $k \in \cal{S}_d$.

For the last item, recall that W_k is associated with the vertices $(s_k,\,t_k (1),\,t_k (2), \ldots,\,t_k (D_k))$, so we are considering some subset $\hat{\cal V}_k$ of the vertices t _k (i), i=1, 2,舰, D _k . We write the corresponding subset of estimates as $\hat W_k (\hat {\cal V}_k)$.

We continue by noting that, for reliable communication, Fano's inequality [24, p. 39] requires that

$$\begin{array}{l}\sum\limits_{k \in \cal{S}_d } {R_k \le \sum\limits_{k \in \cal{S}_d } {\frac{1}{N}I(W_k;\hat W_k (\hat {\cal V}_k))} } \\\quad \quad=\sum\limits_{k=1}^{|\cal{S}_d |} {\frac{1}{N}I\big(W_{\pi (k)};\hat W_{\pi (k)} \big(\hat {\cal V}_{\pi (k)} \big)\big)} . \\\end{array}$$

(A.1)

We define $W_\pi ^{k - 1}=[W_{\pi (1)},W_{\pi (2)}, \ldots,W_{\pi (k - 1)} ]$ and bound

$$\begin{array}{l}I\big(W_{\pi (k)};\hat W_{\pi (k)} \big(\hat {\cal V}_{\pi (k)} \big)\big) \\({\rm a}) \\\le I\big(W_{\pi (k)};\hat W_{\pi (k)} \big(\hat {\cal V}_{\pi (k)} \big)Y_{\varepsilon_d }^N Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } W_\pi ^{k - 1} \big) \\({\rm b}) \\= I\big(W_{\pi (k)};\hat W_{\pi (k)} \big(\hat {\cal V}_{\pi (k)} \big)Y_{\varepsilon_d }^N |Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } W_\pi ^{k - 1} \big) \\({\rm c}) \\= I\big(W_{\pi (k)};Y_{\varepsilon_d }^N |Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } W_\pi ^{k - 1} \big) \\\end{array}$$

(A.2)

where (a) follows because $I(A;B) \le I(A;BC)$, (b) follows because the messages and noise are statistically independent, and (c) follows by the chain rule for mutual information and because success in step 2) in Section 5 implies that

$$I\big(W_{\pi (k)};\hat W_{\pi (k)} \big(\hat {\cal V}_{\pi (k)} \big)|Y_{\varepsilon_d }^N Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } W_\pi ^{k - 1} \big)=0$$

(A.3)

via fd-separation. Inserting (A.2) into (A.1) and applying the chain rule for mutual information, we find that

$$\sum\limits_{k \in \cal{S}_d } {R_k \le \frac{1}{N}I\big(W_{\cal{S}_d };Y_{\varepsilon_d }^N |Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } \big).}$$

(A.4)

We continue by upper bounding the mutual information expression in (A.4) by

$$\begin{array}{*{20}l}{I\big(W_{\cal{S}_d };Y_{\varepsilon_d }^N |Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } \big)} \hfill \\({\underline{\underline {{\rm a}}}) \sum\limits_{n=1}^N {I\big(W_{\cal{S}_d };Y_{\varepsilon_d }^{(n)} |Y_{\varepsilon_d }^{n - 1} Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } \big)} } \hfill \\{\le \sum\limits_{n=1}^N {I\big(W_{\cal{S}_d } X_{\varepsilon_d }^{(n)};Y_{\varepsilon_d }^{(n)} |Y_{\varepsilon_d }^{n - 1} Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } \big)} } \hfill \\({\underline{\underline {{\rm b}}}) \sum\limits_{n=1}^N {\left[ {H\big(Y_{\varepsilon_d }^{(n)} |Y_{\varepsilon_d }^{n - 1} Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } \big) - H\big(Y_{\varepsilon_d }^{(n)} |X_{\varepsilon_d }^{(n)} \big)} \right]} } \hfill \\({\underline{\underline {{\rm c}}}) \sum\limits_{n=1}^N {\left[ {H\big(Y_{\varepsilon_d }^{(n)} |Y_{\varepsilon_d }^{n - 1} Z_{\varepsilon_d^C }^N W_{\cal{S}_d^C } \big) - \sum\limits_{e \in \varepsilon_d } {H\big(Y_{e}^{(n)} |X_{e}^{(n)} \big)} } \right]} } \hfill \\{({\rm d}) \le \sum\limits_{n=1}^N {\sum\limits_{e \in \varepsilon_d } {\left[ {H\big(Y_{e}^{(n)} - H\big(Y_e^{(n)} |X_e^{(n)} \big)} \right]} } } \hfill \\{\le \sum\limits_{e \in \varepsilon_d } {\sum\limits_{n=1}^N {\max\limits_{P_{X_e^{(n)}} } I\big(X_e^{(n)};Y_e^{(n)} \big)} } } \hfill \\({\underline{\underline {e}}) \sum\limits_{e \in \varepsilon_d } {N \cdot C_e } } \hfill \\\end{array}$$

(A.5)

where (a) follows by the chain rule for mutual information, (b) and (c) follow by (3.1), (d) follows because conditioning cannot increase entropy, and (e) follows because it is known that (see [24], Ch. 8)

$$C_e=\mathop {\max }\limits_{P_{X_e^{(n)} } } I\big(X_e^{(n)};Y_e^{(n)} \big)$$

(A.6)

Inserting (A.5) into (A.4) gives (5.1).