### Abstract

We consider an algorithmic model of diffusion networks, in which n nodes are distributed in a 2D Euclidean space and communicate by diffusing and sensing molecules. Such a model is interesting on its own right, although from the distributed computing point of view it may be seen as a generalization or even a framework for other wireless communication models, such as the SINR model, radio networks or the beeping model. Additionally, the diffusion networks model formalizes and generalizes recent case studies of simple processes in environment where nodes, often understood as biological cells, communicate by diffusing and sensing simple chemical molecules. To demonstrate the algorithmic nature of our model, we consider a fundamental problem of reaching consensus by nodes: in the beginning each node has some initial value, e.g., the reading from its sensor, and the goal is that each node outputs the same value. Our deterministic distributed algorithm runs at every node and outputs the consensus value equal to the sum of inputs divided by the sum of the channel coefficients of each cells. For a node v consensus is reached in (formula presented) communication rounds, where d_{v} is the sum of molecule reachability ratios to node v in the medium, d_{min}=min_{i}d_{i}, d_{max}=max_{i}d_{i} and b is the sum of the initial values. p represents the second largest eigenvalue of a matrix of normalised molecule reachability ratios, that we analyze together with an associated Markov Chain.

Original language | English (US) |
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Title of host publication | Algorithms for Sensor Systems - 14th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2018, Revised Selected Papers |

Editors | Seth Gilbert, Danny Hughes, Bhaskar Krishnamachari |

Publisher | Springer Verlag |

Pages | 180-192 |

Number of pages | 13 |

ISBN (Print) | 9783030140939 |

DOIs | |

State | Published - Jan 1 2019 |

Externally published | Yes |

Event | 14th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2018 - Helsinki, Finland Duration: Aug 23 2018 → Aug 24 2018 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11410 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2018 |
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Country | Finland |

City | Helsinki |

Period | 8/23/18 → 8/24/18 |

### Fingerprint

### Keywords

- Ad hoc networks
- Cells
- Consensus
- Diffusion networks
- Molecules

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms for Sensor Systems - 14th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2018, Revised Selected Papers*(pp. 180-192). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11410 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-14094-6_12

**Reaching Consensus in Ad-Hoc Diffusion Networks.** / Kowalski, Dariusz R.; Mirek, Jarosław.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms for Sensor Systems - 14th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2018, Revised Selected Papers.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11410 LNCS, Springer Verlag, pp. 180-192, 14th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2018, Helsinki, Finland, 8/23/18. https://doi.org/10.1007/978-3-030-14094-6_12

}

TY - GEN

T1 - Reaching Consensus in Ad-Hoc Diffusion Networks

AU - Kowalski, Dariusz R.

AU - Mirek, Jarosław

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We consider an algorithmic model of diffusion networks, in which n nodes are distributed in a 2D Euclidean space and communicate by diffusing and sensing molecules. Such a model is interesting on its own right, although from the distributed computing point of view it may be seen as a generalization or even a framework for other wireless communication models, such as the SINR model, radio networks or the beeping model. Additionally, the diffusion networks model formalizes and generalizes recent case studies of simple processes in environment where nodes, often understood as biological cells, communicate by diffusing and sensing simple chemical molecules. To demonstrate the algorithmic nature of our model, we consider a fundamental problem of reaching consensus by nodes: in the beginning each node has some initial value, e.g., the reading from its sensor, and the goal is that each node outputs the same value. Our deterministic distributed algorithm runs at every node and outputs the consensus value equal to the sum of inputs divided by the sum of the channel coefficients of each cells. For a node v consensus is reached in (formula presented) communication rounds, where dv is the sum of molecule reachability ratios to node v in the medium, dmin=minidi, dmax=maxidi and b is the sum of the initial values. p represents the second largest eigenvalue of a matrix of normalised molecule reachability ratios, that we analyze together with an associated Markov Chain.

AB - We consider an algorithmic model of diffusion networks, in which n nodes are distributed in a 2D Euclidean space and communicate by diffusing and sensing molecules. Such a model is interesting on its own right, although from the distributed computing point of view it may be seen as a generalization or even a framework for other wireless communication models, such as the SINR model, radio networks or the beeping model. Additionally, the diffusion networks model formalizes and generalizes recent case studies of simple processes in environment where nodes, often understood as biological cells, communicate by diffusing and sensing simple chemical molecules. To demonstrate the algorithmic nature of our model, we consider a fundamental problem of reaching consensus by nodes: in the beginning each node has some initial value, e.g., the reading from its sensor, and the goal is that each node outputs the same value. Our deterministic distributed algorithm runs at every node and outputs the consensus value equal to the sum of inputs divided by the sum of the channel coefficients of each cells. For a node v consensus is reached in (formula presented) communication rounds, where dv is the sum of molecule reachability ratios to node v in the medium, dmin=minidi, dmax=maxidi and b is the sum of the initial values. p represents the second largest eigenvalue of a matrix of normalised molecule reachability ratios, that we analyze together with an associated Markov Chain.

KW - Ad hoc networks

KW - Cells

KW - Consensus

KW - Diffusion networks

KW - Molecules

UR - http://www.scopus.com/inward/record.url?scp=85063420037&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063420037&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-14094-6_12

DO - 10.1007/978-3-030-14094-6_12

M3 - Conference contribution

AN - SCOPUS:85063420037

SN - 9783030140939

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 180

EP - 192

BT - Algorithms for Sensor Systems - 14th International Symposium on Algorithms and Experiments for Wireless Sensor Networks, ALGOSENSORS 2018, Revised Selected Papers

A2 - Gilbert, Seth

A2 - Hughes, Danny

A2 - Krishnamachari, Bhaskar

PB - Springer Verlag

ER -