If you were surprised at the success of the LinkedIn (LNKD) initial public offering in May, you weren’t alone. The company’s management thought it might be worth around $3 billion, according to advance filings with the Securities & Exchange Commission. The marketplace saw differently and valued the company at more than $9 billion in the first week of trading—even though LinkedIn does not expect to make a profit in 2011. Does this soaring stock price signal the next dot-com boom? Hardly. LinkedIn’s success lies in its ability to provide what I call “the power of weak connections.”

The classic prescription for success in business was built on the premise that “it’s not what you know, it’s who you know.” You should make a few powerful friends and cultivate those relationships assiduously. Play golf where the chief executive officer plays golf, arrange for “accidental” encounters, and be prepared to dazzle.

The power of weak connections rests on a very different assumption: Don’t worry so much about having a small number of close friends. Instead, concentrate on making a much larger number of acquaintances—who might be called “not-so-close friends.” That, of course, is where LinkedIn and social media tools like it come in.

I think there is something deep in human nature that encourages us to both help and seek help from one another. This aid circle comprises our family members, loved ones, and closest friends. It also seems to include almost anyone with whom we have some sort of connection, no matter how weak. Even the most tenuous of links—a connection through an acquaintance of an acquaintance on LinkedIn or Facebook—suffices to overcome the countervailing primal instinct: fear of strangers.

You could make the case that a host of fraternal organizations such as the Freemasons and alumni groups are also sustained by our need to forge weak connections with groups. What is new is the technology that allows us to do this, remotely and asynchronously. Today we just have to make some clicks happen and hope that the other person responds with a “yes” to forge a new connection.

How H.O.G. Helped Save Harley-Davidson

Businesses can derive enormous benefits from creating communities that are based on weak connections among members. My favorite example is the Harley Owners Group, or H.O.G. When H.O.G. began in 1983, Milwaukee-based Harley-Davidson (HOG) was the last U.S. motorcycle manufacturer putting new models on the road. The company had massive debt in the wake of a leveraged buyout and was losing market share to the Japanese. Disaffected dealers banded together in 1983 and approached the company’s lead lender, Citicorp (C), in hopes of putting pressure on management to improve product quality and provide better service to core U.S. dealers.

Amid so much turmoil, and mainly to placate the angry dealers, Harley created H.O.G. The structure was simple: Each dealership had the opportunity to sponsor a local H.O.G. chapter. (None would be sanctioned without a dealership behind it.) Aside from basic support from Milwaukee, the chapters were effectively volunteer organizations that would charge membership dues to support local activities—mostly members-only rides. H.O.G. was an immediate hit. Today, more than a million brothers and sisters around the world are “bound by the passion to ride,” according to the company.

I can point to four lessons that “offline” businesses can take away from LinkedIn’s and H.O.G.’s use of weak connections:

1. Offer a glorious product or service. A great product makes its own friends. Catalina Yachts, Porsche (PAH3) cars, Apple (AAPL) computers, and Leica Camera (LCA1) devices are all examples of products with loyal and powerful user groups.

2. Think “customer to customer.” Most businesses are stuck in the traditional mindset of building relationships on a hub-and-spoke model, with the company being the hub. This is only a tiny part of today’s community-building game. Udi’s Gluten Free Foods has built a community of some 200,000 Facebook fans because customers want to connect with other people facing the challenges of gluten intolerance.

3. Reach noncustomers, too. If you can use the Internet to provide something valuable to people who aren’t currently your customers, you may convert them—especially if your customers are also communicating with them. A high-end tree-maintenance company in my town has a website that’s full of information that changes on a weekly basis during the growing season: when and how to fertilize, mulch, prune, and so forth. When you decide that you’re not going to do all that landscaping yourself, who are you going to call?

4. Embrace weak connection technologies. Even if you sell a very conventional product—be it shoes, sausages, or screwdrivers—you can capture value from weak connections by leveraging the power of today’s digital networking tools. It is not uncommon for forward-thinking businesses with under $100 million in sales to spend $100,000 a month on social-networking tools. Retailers and consumer packaged-goods companies, as well as organizations such as IBM (IBM) and Dell (DELL), have created entire departments focused on listening and responding to what consumers and business executives are talking about online. These companies also invest in outside agencies and tools to facilitate search engine optimization (SEO), monitor brand mentions, and develop online campaigns involving Twitter and Facebook. These programs are paying off. In his book, Real Time Marketing & PR, David Meerman Scott recounts a powerful example of how a quick response by phone-equipment maker Avaya to a question posed on Twitter resulted in a $250,000 sale.

C. S. Lewis said it well. “Friendship is born at that moment when one person says to another: ‘What! You too? I thought I was the only one!’” Savvy businesses can draw on this insight by using the power of weak connections to build community and create value.