cut vertex – Pertinent Observations

When I’m visiting someone’s house and they have an accessible bookshelf, one of the things I do is to go check out the books they have. There is no particular motivation, but it’s just become a habit. Sometimes it serves as conversation starters (or digressors). Sometimes it helps me understand them better. Most of the time it’s just entertaining.

So at a friend’s party last night, I found this book on Graph Theory. I just asked my hosts whose book it was, got the answer and put it back.

As many of you know, whenever we host a party, we use graph theory to prepare the guest list. My learning from last night’s party, though, is that you should not only use graph theory to decide WHO to invite, but also to adjust the times you tell people so that the party has the best outcome possible for most people.

With the full benefit of hindsight, the social network at last night’s party looked approximately like this. Rather, this is my interpretation of the social network based on my knowledge of people’s affiliation networks.

This is approximate, and I’ve collapsed each family to one dot. Basically it was one very large clique, and two or three other families (I told you this was approximate) who were largely only known to the hosts. We were one of the families that were not part of the large clique.

This was not the first such party I was attending, btw. I remember this other party from 2018 or so which was almost identical in terms of the social network – one very large clique, and then a handful of families only known to the hosts. In fact, as it happens, the large clique from the 2018 party and from yesterday’s party were from the same affiliation network, but that is only a coincidence.

Thinking about it, we ended up rather enjoying ourselves at last night’s party. I remember getting comfortable fairly quickly, and that mood carrying on through the evening. Conversations were mostly fun, and I found myself connecting adequately with most other guests. There was no need to get drunk. As we drove back peacefully in the night, my wife and daughter echoed my sentiments about the party – they had enjoyed themselves as well.

This was in marked contrast with the 2018 party with the largely similar social network structure (and dominant affiliation network). There we had found ourselves rather disconnected, unable to make conversation with anyone. Again, all three of us had felt similarly. So what was different yesterday compared to the 2018 party?

I think it had to do with the order of arrival. Yesterday, we were the second family to arrive at the party, and from a strict affiliation group perspective, the family that had preceded us at the party wasn’t part of the large clique affiliation network (though they knew most of the clique from beforehand). In that sense, we started the party on an equal footing – us, the hosts and this other family, with no subgroup dominating.

The conversation had already started flowing among the adults (the kids were in a separate room) when the next set of guests (some of them from the large clique arrived), and the assimilation was seamless. Soon everyone else arrived as well.

The point I’m trying to make here is that because the non-large-clique guests had arrived first, they had had a chance to settle into the party before the clique came in. This meant that they (non-clique) had had a chance to settle down without letting the party get too cliquey. That worked out brilliantly.

In contrast, in the 2018 party, we had ended up going rather late which meant that the clique was already in action, and a lot of the conversation had been clique-specific. This meant that we had struggled to fit in and never really settled, and just went through the motions and returned.

I’m reminded of another party WE had hosted back in 2012, where there was a large clique and a small clique. The small clique had arrived first, and by the theory in this post, should have assimilated well into the party. However, as the large clique came in, the small clique had sort of ended up withdrawing into itself, and I remember having had to make an effort to balance the conversation between all guests, and it not being particularly stress-free for me.

The difference there was that there were TWO cliques with me as cut-vertex. Yesterday, if you took out the hosts (cut-vertex), you would largely have one large clique and a few isolated nodes. And the isolated nodes coming in first meant they assimilated both with one another and with the party overall, and the party went well!

And now that I’ve figured out this principle, I might break my head further at the next party I host – in terms of what time I tell to different guests!

When I’m meeting a random bunch of people I like to graph out social networks in terms of who knew whom before the meeting happened. For example, I was meeting some friends yesterday – B was in town, and wanted to meet people. He called A and C, who got along D (also known to B). After this meeting B was supposed to meet E, but E landed up anyhow. Based on who knew whom before the meeting this is how the network topology went. People are represented by vertices and if there’s an edge between them that means they know each other.

So it started with A and B meeting, with C supposed to land up in a while. Now, C knows A and B through two different “affiliation groups”, but knows both quite well. So C lands up, but now the question is what do you talk about. The basic structure of the group – where A-B, B-C and C-A know each other through three separate affiliation groups means you can’t talk about people (thankfully!).

Anyway conversation goes on, and then D lands up. When B asked C if they could meet, he said “I’m not in touch with anyone else here in Bangalore. But if you think there’s someone else from our affiliation group who’s here and wants to meet, bring them along”. Thus, C invites D (whom he hasn’t met for ages) and D lands up.

Now, for the first time, the group is not a clique – since A and D don’t know each other. It’s up to B and C now to control the conversation in a way that A or D don’t get bored. People talk about work, careers and all that – where anyone can give gyaan.

After a while, E lands up. Now, E doesn’t know anyone else in the group (apart from B). So now, B becomes a cut-vertex. B starts talking to E. With B and E taken out, in the A-C-D network, C is now a cut-vertex! So it’s up to C to manage the conversation with A and D! C isn’t particularly good at that!

Soon A leaves. Now, the group effectively splits, while sitting at the same table. B talks to E (no one else knows E), and C talks to D. All is well.

The problem with the group was that none of the “connectors” (B, C) were particularly good at connecting people, and keeping one conversation. This, though, wasn’t the case at a drinks session I attended on Monday evening. There, the social network at the beginning of the conversation looked like this (variables here all mean different people, only I was common to both meetings):

The thin lines here indicate that B-F and E-F had met before, but didn’t know each other well enough. As you can see, A is now the cut-vertex here. The difference, though, is that A is a master networker, and has a self-professed interest in “collecting interesting people”. The group for the meeting was also fully curated by A – no one “brought along” anyone else.

So A ensured that the conversation flowed. He made sure people connected, and there was great conversation. At the end of the day the network was a clique!

I’ve never been good at making these connections. I dread gathering where I’m the cut-vertex – forever afraid that someone might be left out. Connecting and collecting people is surely a skill I need to develop!

PS: At a coffee shop in Mumbai eight summers ago, I was at one end of the social network which looked like this. Don’t ask me how it came about!