By Herbert Lash

NEW YORK (Reuters) - WeWork Cos, the office-space sharing start-up with an estimated $16 billion valuation, has demanded that one of its tenants take down a blog on a public website that suggests the rate at which members leave has surged in recent months.

The posting on Tuesday by Justin Zhen, whose data analytics firm Thinknum operates out of a WeWork site on 42nd Street in Manhattan, also said that member engagement on WeWork's social networking platform is surprisingly low.

The WeWork platform is seen inside the company as a key tool for creating a sense of community at WeWork locations. The company sees its fortunes tied to the premise that the entrepreneurs and young urban workers populating its workspaces will form lasting communities in part by being active on the network.

WeWork is a rare "decacorn," a start-up valued at $10 billion or more.

Zhen's analysis said WeWork's historical churn rate of 1.2 percent annually had shot up in recent months to 6 percent. The churn rate is a ratio of cancellations to total WeWork members.

The analysis also said more than three-quarters of WeWork's members had never posted on the network.

WeWork officials dispute Zhen's points and said his data is inaccurate or incomplete.

"This company scraped incomplete data from our member network, which violated the rules of our community," WeWork said in a statement to Reuters. "The data they published and their associated conclusions are factually inaccurate and do not reflect our thriving business."

More than 80 percent of WeWork members actively use the company's digital services each month, the company said.

Lawyers for WeWork gave Thinknum until Saturday to remove the blog on online publisher medium.com and to delete all the data the firm had gathered from the network, which contains e-mail addresses, telephone numbers and dates for when members joined and left.

A cease-and-desist letter sent to Zhen on Wednesday and viewed by Reuters said Thinknum agreed when becoming a member in April 2015 that it would not engage in scraping, spidering, crawling or using any other technology or software to access WeWork data.

Thinknum also agreed not to collect e-mail addresses or other contact information, according to the letter signed by Douglas Curtis, a deputy general counsel at WeWork.

Zhen said he does not believe he violated the terms of his WeWork membership when he used an application programming interface, or API, that he and his Thinknum colleagues found on the WeWork network to locate and analyze WeWork membership data. He said he was surprised the network, which contains member telephone numbers and e-mail addresses, among other details, was so accessible.

"We couldn't believe the data was out there like that," Zhen said. "You shouldn't do this if you're a big company."

WeWork said to Reuters the data Thinknum gathered was incomplete as some of its biggest customers restrict access to the network, and that certain profiles were created for testing purposes. In addition, Thinknum does not have access to certain data that better reflects a network that is thriving, WeWork said.

Zhen's blog comes on the heels of a restraining order WeWork sought on Monday against a former employee it accused of disclosing unauthorized information. The disclosure, which showed WeWork had missed certain financial targets, caused the company "significant damage," it said in a lawsuit it filed against the ex-employee in New York State Supreme Court in Manhattan last week.

Thinknum said it created its analysis by tapping the software application that governs the network, which is used by members to communicate among themselves and with WeWork employees.

Thinknum was founded in 2013 by Zhen and his former Princeton University roommate Gregory Ugwi. The firm supplies data about publicly traded companies to hedge funds and banks, including Goldman Sachs, which cited Thinknum in May in a research note showing loan origination by Lending Club Corp. was in decline.

The company, which Zhen said is now profitable, employs 10 people, including six working at the WeWork location in mid-town Manhattan.

(Reporting by Herbert Lash; Editing by Dan Burns and Andrew Hay)