# A Review of Advanced Algebraic Approaches Enabling Network Tomography for Future Network Infrastructures

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## Abstract

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## 1. Introduction

`traceroute`and

`ping`, which require Internet Control Message Protocol (ICMP) responses from routers. Furthermore, this approach avoids the current problem of an increasing number of Internet routers blocking

`traceroute`requests due to privacy and security concerns (such routers are referred to as anonymous routers).

## 2. Network Tomography Model and Classification Overview

#### 2.1. Linear Measurement Model and Basic Definitions

**Assumption**

**1**(Spatial Independence)

**.**

**Assumption**

**2**(Temporal Stationarity)

**.**

#### 2.2. Network Tomography Classification Overview

- Link-level parameter estimation: vector ${\mathit{Y}}_{\mathit{t}}$ consists of end-to-end traffic measurements such as accounts of delivered/lost packets and time delays (differences) between packet transmissions and receptions. The unknown variables (i.e., vector ${\mathit{X}}_{\mathit{t}}$) are the link metrics, which are typically additive, meaning that the path metric obtained by combining multiple serial links is the sum of the individual link metrics. Time delays are a good example of an additive metric, whereas a multiplicative metric, such as loss rates, can be expressed in an additive form by using the logarithmic function. Since these end-to-end path measurements are carried out between pairs of nodes with monitoring capabilities, the decision of which nodes will be chosen as monitors becomes a key problem. This category can be further divided based on the different application contexts and the specific link metric used (e.g., loss tomography [8,9,10], delay tomography [11,12,13], bandwidth tomography [14], etc.)
- Path-level traffic intensity estimation [15,16,17]: the goal is to estimate the traffic volume (or to infer the distribution of the flowing traffic) of end-to-end routes between all pairs of nodes in the network based on the observed traffic volumes on individual links. At a practical level, the involved link aggregate data are acquired by monitoring the total number of packets traversing the respective nodes. The combination of the traffic intensities of all these origin–destination pairs forms the origin–destination (OD) traffic matrix. Hence, this category aims at the inference of OD traffic matrices (${\mathit{X}}_{\mathit{t}}$) based on (aggregated) link traffic counts (${\mathit{Y}}_{\mathit{t}}$). This is the original NT problem studied in [18].
- Topology inference [19,20,21,22,23]: the network topology, which is expressed by the routing matrix $\mathit{A}$, is unknown. Hence, in this category, the goal is the inference of the topology based on end-to-end measurements conducted at the network edge and obtained without the cooperation and the participation of the internal nodes. These measurements evaluate the degree of correlation between receivers. Network topology is usually examined in terms of the logical topology, which is defined by the branching points between paths to different destination nodes. Internal nodes where no branching of traffic occurs do not appear in the logical topology. The central idea of the methods in this category is to obtain a measure of similarity between pairs of terminal nodes (i.e., source and destination) by means of end-to-end measurements that behave as a monotonically increasing function of the number of shared links or common queues between the two receiving nodes. Knowledge of the pairwise similarity metric values under an additive metric is sufficient to completely identify the logical topology by employing various statistical techniques such as hierarchical clustering, maximum likelihood, and Bayesian inference.

## 3. Network Coding Enabled Network Tomography

#### 3.1. Topology Inference

#### 3.2. Loss Estimation

## 4. Compressed Sensing Enabled Network Tomography

- enable accurate reconstruction of ${\mathit{X}}_{\mathit{t}}$ from ${\mathit{Y}}_{\mathit{t}}$ when ${\mathit{X}}_{\mathit{t}}$ is known to be sparse;
- use fast decoding algorithm; and
- make the least possible measurements (maximum achievable compression of $\mathit{A}$) [41].

#### 4.1. Delay Estimation

#### 4.2. Loss Estimation

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NT | Network Tomography |

QoS | Quality of Service |

ICMP | Internet Control Message Protocol |

EM | Expectation-Maximization |

MCMC | Markov Chain Monte Carlo |

MoM | Method of Moments |

NC | Network Coding |

CS | Compressed Sensing |

OD | Origin-Destination |

RLNC | Random Linear Network Coding |

XOR | Exclusive Or |

IRV | Impulse Response Vector |

NRSC | Network Reed-Solomon Coding |

DAG | Directed Acyclic Graph |

SCFS | Smallest Consistent Failure Set |

LA | Linear Algebraic |

LA-NE | Linear Algebraic with Normal Equations |

LA-RS | Linear Algebraic with Row Selection |

BP | Belief Propagation |

WSN | Wireless Sensor Network |

MLE | Maximum Likelihood Estimator |

LP | Linear Programming |

i.i.d. | independent and identically distributed |

MSE | Mean Squared Error |

MT | Multicast Tree |

RMT | Reversed Multicast Tree |

MAP | Maximum A-Posteriori |

WSCP | Weighted Set Cover Problem |

GFT | Graph Fourier Transform |

BS | Base Station |

RTT | Round Trip Time |

CDS | Connected Dominating Set |

RIP | Restricted Isometry Property |

MIS | Maximal Independent Set |

FPR | False Positive Rate |

DR | Detection Rate |

RMSE | Root Mean Square Error |

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Network Coding | Compressed Sensing | Loss | Delay | Topology | Comments | |
---|---|---|---|---|---|---|

Fragouli et al. [27] | ✓ | ✓ | binary trees | |||

Sattari et al. [28] | ✓ | ✓ | m-ary trees, M-by-N DAGs | |||

Jithin et al. [30] | ✓ | ✓ | M-by-N DAGs, asynchronous sources | |||

Yao et al. [31] | ✓ | ✓ | RLNC, NRSC, passive tomography | |||

Mohammad et al. [33] | ✓ | RLNC, congested link location | ||||

Gui et al. [34] | ✓ | ✓ | mesh DAG topologies | |||

Gui et al. [35] | ✓ | ✓ | mesh DAG topologies, minimum probe size | |||

Shah-Mansouri et al. [36] | ✓ | ✓ | WSN, RLNC, subspace property, virtual sources | |||

Sattari et al. [37] | ✓ | ✓ | trees and general topologies | |||

Sattari et al. [38] | ✓ | ✓ | trees with multiple sources | |||

Fan et al. [52] | ✓ | ✓ | weighted ${\ell}_{1}$ minimization, expander graphs | |||

Chen et al. [53] | ✓ | ✓ | link congestion probabilities, greedy iterative algorithm | |||

Takemoto et al. [54] | ✓ | ✓ | measurement paths construction, low-quality link detection | |||

Morita et al. [56] | ✓ | ✓ | wireless multihop networks, GFT, spatially dependent channels | |||

Bandara et al. [57] | ✓ | ✓ | scalable, adaptive fault localization | |||

Firooz et al. [41] | ✓ | ✓ | k-identifiability, expander graphs | |||

Fattaholmanan et al. [43] | ✓ | collaborative distributed framework, individual matrix for every node | ||||

Wang et al. [44] | ✓ | ✓ | expander graphs, ${\ell}_{1}$ minimization | |||

Fan et al. [45] | ✓ | ✓ | synchronization errors, constrained ${\ell}_{1}-{\ell}_{2}$ optimization | |||

Nakanishi et al. [46] | ✓ | ✓ | no clock synchronization, reflective NT | |||

Nakanishi et al. [47] | ✓ | ✓ | synchronization errors, differential routing matrix | |||

Kinsho et al. [48] | ✓ | ✓ | mobile networks, GFT & passive measurements | |||

Wei et al. [49] | ✓ | ✓ | dynamic networks, line graph model |

Description of Key Points, Advantages, and Concise Performance Evaluation | |
---|---|

Fragouli et al. [27] | deterministic topology recovery in one pass for binary tree networks without packet losses; rapid topology inference after only a few probes in the presence of packet losses; the probability of error is increasing with the loss rates of the links and decreasing with the number of probes per iteration |

Sattari et al. [28] | extends [27] to (full or general) m-ary undirected trees; accurately distinguishes among all different 2-by-2 components for DAGs; more accurate with less experiments compared to [29]; smaller complexity and less bandwidth consumption than [31] |

Jithin et al. [30] | deterministically recovers the exact DAG logical topology with a single probing experiment and one probe packet traversing each link; contrary to [28,29], no synchronization among sources is required |

Yao et al. [31] | algorithms with polynomial time computational complexity in the network size for topology inference under random errors and failure localization under random and adversarial errors; adversarial error localization under NRSC does not require prior knowledge of the topology |

Mohammad et al. [33] | sufficient conditions for successfully identifying a single congested link in any logical network; speed (length of training sequence)/complexity (size of NC packets) trade-off |

Gui et al. [34] | lower bound on probe size for valid end-to-end observations under NC; consistent estimators of link loss rates based on LA approach; LA achieves better accuracy (lower RMSE) than BP for $n>400$ |

Gui et al. [35] | extends [34]; LA using Method of Row Selection (LA-RS) has lower complexity but similar RMSE compared to LA using Method of Normal Equations (LA-NE); bandwidth consumption/estimation accuracy trade-off; LA-RS is more accurate than the BP approach; RMSE decreases as the average link success rate increases; a graph with more sources achieves better estimation accuracy; RMSE linearly increases with the number of nodes |

Sattari et al. [38] | proposes a low complexity algorithm to compute MLEs of link loss rates in multiple-source tree networks with multicast and NC capabilities; the algorithm is efficient as the computation at each node is at worst proportional to the tree depth |

Sattari et al. [37] | extends the work in [38]; NC loss tomography is more efficient than traditional methods without NC; minimum cost routing in polynomial time using NC; BP (applies on trees and on general topologies) versus MINC-like heuristics (introduced in [8] only apply on trees) performance depends on number of sources (the more sources the better for both); the larger the loss rates, the slower the convergence of BP algorithm; BP performance increases as n increases |

Fan et al. [52] | CS with weighted ${\ell}_{1}$ minimization; weights are determined based on the prior congestion probabilities of links; outperforms (unweighted) ${\ell}_{1}$ minimization in terms of estimation accuracy, DR, and FPR |

Chen et al. [53] | greedy iterative estimation algorithm to locate congested links based on prior probabilities; recovered probability is not very accurate; DR is high, whereas FPR not very good; performance increases while congestion ratio decreases |

Takemoto et al. [54] | window-based sequential loss tomography scheme for networks with dynamically changing link loss rates; increases the sparsity of link loss rates by subtracting them from steady-state path loss rates; path construction algorithm between two monitors that maximizes identifiability |

Morita et al. [56] | estimates the GFT of a node state vector and uses CS to estimate packet loss rates in wireless networks with spatially dependent channels; spatial dependence does not imply strong correlation between neighboring nodes’ loss rates; limited applicability |

Bandara et al. [57] | monitors network for anomaly detection and localizes faulty links using CS; scalable $\mathcal{O}\left(\mathrm{log}\mathcal{E}\right)$; cost-effective |

Fan et al. [45] | link delays as exponential random variables; additive Gaussian clock synchronization error; constrained ${\ell}_{1}-{\ell}_{2}$ optimization that outperforms the original one [60] in estimation error and detection probability |

Nakanishi et al. [46] | reflective NT with a single transceiver for identifying bottleneck links; efficient algorithm for constructing reflective (fully-loop and folded) paths; no clock synchronization required |

Nakanishi et al. [47] | synchronization-free delay tomography; differential routing matrix construction by selecting a path as reference; comparable performance to conventional delay schemes |

Kinsho et al. [48] | heterogeneous CS-based delay tomography in mobile networks; crowd-sourcing passive RTT measurements; estimates delays at BSs and at servers from the GFT of the vector of the spatial-dependent delays at BSs; the scheme captures the spatial dependence of delays and is robust to the presence of inactive BSs |

Wei et al. [49] | extends [50] to link delay estimation with dynamic link operations (i.e., insertion and deletion) and provides an algorithm with better theoretical upper bound on running time; topological constraints represented by a line graph model |

Error (RMSE) | Entropy | FPR | DR | |
---|---|---|---|---|

LA-NE [34] | [0.5%, 14.0%) | |||

LA-RS [35] | [0.5%, 15.0%) | |||

BP [37] | [2.0%, 15.0%) | (−320.0, −180.0) ${}^{*}$ | ||

(−400.0, −230.0) ${}^{\u2020}$ | ||||

MINC-like heuristic [37] | (−350.0, −180.0) ${}^{*}$ | |||

(−360.0, −200.0) ${}^{\u2020}$ | ||||

Fan et al. [52] | (20.0%, 30.0%] ${}^{**}$ | (2.0%, 8.0%) | (75.0%, 100.0%] | |

Chen et al. [53] | (10.0%, 30.0%] | (70.0%, 100.0%) | ||

Takemoto et al. [54] | (99.5%, 100.0%) | |||

Bandara et al. [57] | [0, 1.0%] | [99.0%, 100.0%] | ||

Fan et al. [45] | [2.4%, 3.5%] ${}^{\u2021}$ | [0%, 12.0%) | [40.0%, 89.0%) | |

Nakanishi et al. [46] | [18.0%, 100.0%] | |||

Kinsho et al. [48] | <20.0% |

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**MDPI and ACS Style**

Kakkavas, G.; Gkatzioura, D.; Karyotis, V.; Papavassiliou, S.
A Review of Advanced Algebraic Approaches Enabling Network Tomography for Future Network Infrastructures. *Future Internet* **2020**, *12*, 20.
https://doi.org/10.3390/fi12020020

**AMA Style**

Kakkavas G, Gkatzioura D, Karyotis V, Papavassiliou S.
A Review of Advanced Algebraic Approaches Enabling Network Tomography for Future Network Infrastructures. *Future Internet*. 2020; 12(2):20.
https://doi.org/10.3390/fi12020020

**Chicago/Turabian Style**

Kakkavas, Grigorios, Despoina Gkatzioura, Vasileios Karyotis, and Symeon Papavassiliou.
2020. "A Review of Advanced Algebraic Approaches Enabling Network Tomography for Future Network Infrastructures" *Future Internet* 12, no. 2: 20.
https://doi.org/10.3390/fi12020020