Distributed network center and area estimation

1. A wireless sensor network comprising:

a plurality of sensor nodes each comprising:

a signal receiver configured to receive intermediate information from at least one of one or more neighboring nodes of the plurality of sensor nodes;

one or more processors configured to receive the intermediate information and update the intermediate information based on a soft-max approximation function; and

a transmitter configured to send the intermediate information, as updated, to at least one of the one or more neighboring nodes of the plurality of sensor nodes,

wherein:

for each sensor node of the plurality of sensor nodes:

the sensor node stores local location coordinates for the sensor node; and

the sensor node is devoid of receiving location coordinates for any other of the plurality of sensor nodes;

the plurality of sensor nodes are configured to communicate in a distributed manner for a first plurality of iterations until a final iteration of the first plurality of iterations when a predetermined stopping condition is satisfied;

the plurality of sensor nodes are further configured to generate an estimated center of the wireless sensor network based on the intermediate information updated in the final iteration of the first plurality of iterations;

the wireless sensor network is devoid of a fusion center;

the plurality of sensor nodes are further configured to perform generating an estimated radius of the wireless sensor network based on the estimated center of the wireless sensor network using distributed communication among the plurality of sensor nodes for a second plurality of iterations; and

generating the estimated radius of the wireless sensor network comprises:

generating the estimated radius of the wireless sensor network using max consensus across the second plurality of iterations of a distance from each of the plurality of sensor nodes to the estimated center of the wireless sensor network.

2. The wireless sensor network of claim 1 , wherein:

the plurality of sensor nodes are further configured to determine an estimated coverage area of the wireless sensor network based on the estimated radius.

3. The wireless sensor network of claim 1 , wherein:

the plurality of sensor nodes are configured to perform stochastic gradient descent when communicating in the distributed manner;

a single respective sensor node of the plurality of sensor nodes is active at each of a respective one of the first plurality of iterations; and

the intermediate information determined at the single respective sensor node that is active represents an intermediate center estimate of the wireless sensor network.

4. The wireless sensor network of claim 3 , wherein, during an iteration of the first plurality of iterations:

the single respective sensor node of the plurality of sensor nodes that is active receives the intermediate center estimate from a single respective one of the one or more neighboring nodes of the plurality of sensor nodes associated with the single respective sensor node;

the single respective sensor node that is active updates the intermediate center estimate;

the single respective sensor node that is active randomly selects a single respective one of the one or more neighboring nodes of the plurality of sensor nodes as a destination sensor node; and

the single respective sensor node sends the intermediate center estimate, as updated, to the destination sensor node, to cause the single respective sensor node to become inactive, and the destination sensor node to become active.

5. The wireless sensor network of claim 1 , wherein:

the plurality of sensor nodes are configured to perform diffusion adaptation when communicating in the distributed manner; and

during an iteration of the first plurality of iterations:

all of the plurality of sensor nodes are active; and

each active sensor of the plurality of sensor nodes receives the intermediate information from all of the one or more neighboring nodes associated with the active sensor and sends the intermediate information, as updated, to all of the one or more neighboring nodes associated with the active sensor.

6. The wireless sensor network of claim 1 , wherein:

the predetermined stopping condition is based on a delta of updates to the intermediate information being within a predetermined amount across the plurality of sensor nodes of the wireless sensor network for a predetermined amount of iterations.

7. The wireless sensor network of claim 1 , wherein:

the wireless sensor network is two-dimensional; and

the soft-max approximation function is e β{(a i −x i (t)) 2 +(b i −y i (t)) 2 } , where (a i , b i ) are the local location coordinates for sensor node i, β is a design parameter, and (x i (t), y i (t)) is an intermediate center estimate at iteration t.

8. The wireless sensor network of claim 1 , wherein:

each respective set of the one or more neighboring nodes that are associated with each respective one of the plurality of sensor nodes is determined based on a local communication range of each respective one of the plurality of sensor nodes.

9. A method comprising:

communicating in a distributed manner among a plurality of sensor nodes of a wireless sensor network for a first plurality of iterations until a final iteration of the first plurality of iterations when a predetermined stopping condition is satisfied, wherein each of the plurality of sensor nodes comprises a signal receiver configured to receive intermediate information from at least one of one or more neighboring nodes of the plurality of sensor nodes, one or more processors configured to receive the intermediate information and update the intermediate information based on a soft-max approximation function, and a transmitter configured to send the intermediate information, as updated, to at least one of the one or more neighboring nodes of the plurality of sensor nodes;

generating an estimated center of the wireless sensor network based on the intermediate information updated in the final iteration; and

generating an estimated radius of the wireless sensor network based on the estimated center of the wireless sensor network using distributed communication among the plurality of sensor nodes for a second plurality of iterations,

wherein:

for each sensor node of the plurality of sensor nodes:

the sensor node stores local location coordinates for the sensor node; and

the sensor node is devoid of receiving location coordinates for any other of the plurality of sensor nodes;

the wireless sensor network is devoid of a fusion center; and

generating the estimated radius of the wireless sensor network comprises:

generating the estimated radius of the wireless sensor network using max consensus across the second plurality of iterations of a distance from each of the plurality of sensor nodes to the estimated center of the wireless sensor network.

10. The method of claim 9 further comprising:

determining an estimated coverage area of the wireless sensor network based on the estimated radius.

11. The method of claim 9 , wherein:

the plurality of sensor nodes perform stochastic gradient descent when communicating in the distributed manner;

a single respective sensor node of the plurality of sensor nodes is active at each of a respective one of the first plurality of iterations; and

the intermediate information determined at the single respective sensor node that is active represents an intermediate center estimate of the wireless sensor network.

12. The method of claim 11 , wherein, during an iteration of the first plurality of iterations:

the single respective sensor node of the plurality of sensor nodes that is active receives the intermediate center estimate from a single respective one of the one or more neighboring nodes of the plurality of sensor nodes associated with the single respective sensor node;

the single respective sensor node that is active updates the intermediate center estimate;

the single respective sensor node that is active randomly selects a single respective one of the one or more neighboring nodes of the plurality of sensor nodes as a destination sensor node; and

the single respective sensor node sends the intermediate center estimate, as updated, to the destination sensor node, to cause the single respective sensor node to become inactive, and the destination sensor node to become active.

13. The method of claim 9 , wherein:

the plurality of sensor nodes perform diffusion adaptation when communicating in the distributed manner; and

during an iteration of the first plurality of iterations:

all of the plurality of sensor nodes are active; and

each active sensor of the plurality of sensor nodes receives the intermediate information from all of the one or more neighboring nodes associated with the active sensor and sends the intermediate information, as updated, to all of the one or more neighboring nodes associated with the active sensor.

14. The method of claim 9 , wherein:

the predetermined stopping condition is based on a delta of updates to the intermediate information being within a predetermined amount across the plurality of sensor nodes of the wireless sensor network for a predetermined amount of iterations.

15. The method of claim 9 , wherein:

the wireless sensor network is two-dimensional; and

the soft-max approximation function is e β{(a i −x i (t)) 2 +(b i −y i (t)) 2 } , where (a i , b i ) are the local location coordinates for sensor node i, β is a design parameter, and (x i (t), y i (t)) is an intermediate center estimate at iteration t.

16. The method of claim 9 , wherein:

each respective set of the one or more neighboring nodes that are associated with each respective one of the plurality of sensor nodes is determined based on a local communication range of each respective one of the plurality of sensor nodes.