It goes without saying (but I will say it anyway) that the word "network" has many meanings in IT. The Internet revolution of recent history, seems to have popularized one particular interpretation of the term. Namely, "network" as a system in which things - computers mostly - can talk to each other.
Note the egalitarian nature of this definition. All things can talk to all other things. There is no mention of "client/server" in this formulation. It is as if the distinction between clients and servers - a distinction etched into our brains - is layered on top of our understanding of the word "network".
This is odd because, as any TCP/IP programmer knows, the socket programming paradigm on which HTTP sits is fundamentally client/server in nature. Indeed, HTTP itself is utterly client/server in nature. If anything, pure node-to-node networking is built on top of client-to-server networking.
Web servers are passive communication devices. They sit around while clients do all the work, actively seeking to start conversations with the laid back servers they can connect to.
One extreme interpretation of a client/server network has it that there should be a small (possibly only one) server to which everything else connects as a client.
At the other extreme, some hold that there should be no pre-ordained server in a client/server network. All entities on the network should be able to act both as client and as server as the occasion demands. In other words peer-to-peer networking.
On the Web, there is no central server to which everything connects. There is nothing stopping any computer setting itself up as a HTTPserver and waiting for HTTP clients to happen along. Anything can talk to anything else.
Interestingly, although the Web does not impose any restrictions on who, what or where, servers and clients get together, patterns of interconnections are clearly visible. Simply put, certain servers become "hubs" attracting a disproportionate amount of traffic. Think Google. Think Yahoo. Think CNN.
So, although the Web allows anything to connect to anything, in practice, the many connect to the few. Surprisingly few in actual fact. Such hub effects are not limited to the Web. They occur in all branches of endeavor from biological systems to politics. The phenomenon is discussed in the excellent book "Linked: The New Science of Networks"  by Albert-Laszlo Barabasi  which I heartily recommend.
To date, the hubs on the Web have been created by eyeballs. That is to say, they are created as a result of humans browsing and clicking their way through the visible spectrum of Web space with browsers. In a Web services network, there are no eyeballs involved. Where will the hubs be in this new non-visible network? Very good question.
In theory, any Web service can talk to any other Web service - just as any Web browser can talk to any web server. It won't work that way in practice. There will be hubs. Where? I don't know. Doing what? I don't know, but I'm sure they will exist. If you doubt it, read Barabasi's book. I would be very surprised if the book does not change your mind.