Organizational Math: Influence Maps


A perennial problem that people have is when the organization they work for, or with, makes a bad decision. A common refrain when this happens is “why didn’t they listen to me?” In today’s post we examine how to get them to listen to you using a mathematical tool called an influence map. The key mathematics that permits you to build and use influence maps is the directed graph, an example of which appears at the top of the post. The idea is this: objects in the directed graph are actors — people or organizations with power or influence — and the arrows show which way influence flows and, possibly, what type of influence it is.

Influence maps can be applied to an incredible number of different situations. These include your workplace and community.

It turns out that influence mapping is used in many different places. The image at the top of the post is from a blog on figuring out maps of influence in music, a benign use. The linked site also attempts to map influence in several ways — like subway maps — and Occupy Math thinks other users of influence maps should read the post. Influence mapping is touted as a tool to permit women to gain organizational traction without imitating men.  Occupy Math uses the techniques in the linked book in his own department.  These techniques are probably useful for almost anyone.

There is an organization that works on the social justice applications of influence maps. Influence maps can be a tool for social justice when used to map how corporations influence government. There are companies that have tools for building influence maps. As far as Occupy Math can tell, influence maps are a general purpose, broadly applicable tool based on a simple mathematical construct.

Why use influence maps yourself?

Suppose that you want to steer your organization toward making what you think is a good decision. If you wait for the meeting where the decision is to be made, marshaling facts and figures and preparing a good case, you will often be run over and blindsided by someone with an agenda. The cure for this is to talk to people ahead of time, listen to their views, figure out their concerns, and mold your arguments — including which facts and figures you present — to match the opinions and agendas of others. Sure, no problem, except that at this point you need to be someone who has been in the organization for at least a decade with high-end social skills. The purpose of influence mapping is to put, in diagram form, your best information about who influences whom. This will make who you need to talk to a lot clearer.

The parts list for an influence map includes:

  • A list of actors in the decision,
  • An assessment of the links between those actors, including the type of link.

You put your actors in circles (or ovals or squares). The more power an actor has, the bigger the circle. If one actor influences another, you put an arrow. The arrow can point in both directions — this happens when both actors influence one another. The arrows can colored (or styled) to show the type of influence. If one actor has the ability to give orders to another, that might be a black arrow. If an actor listens to another, but need not do what he says, a red arrow might be appropriate. You might have two arrows between two actors, one black, one red.

Working through an example might help. Melissa, Tanya, and Robert work together in a manufacturing business. Melissa manages production, Tanya manages shipping, and they have roughly the same rank in the company. Robert is one of the floor supervisors that keep production moving and he works for Melissa. One of the problems in this workplace is that Melissa and Tanya don’t get along very well — the general manager has assigned them conflicting goals and told them that good performance might get them a chunk of the other’s budget (The G.M. has been reading one of those scary management theory books). Tanya and Melissa also just don’t click as people — but they are both professional enough to put company performance ahead of more petty objectives. Tanya has learned that if she has to send ideas to Melissa that are not viewed as “Tanya’s ideas” she can use Robert as an informational back channel. Robert, who just hates it when production has a problem, is willing to talk with Tanya about any issues he’s having with shipping or anything else. Let’s turn this situation into a (very small) influence map.


Black arrows represent the boss/employee relationship, red arrows represent a willingness to listen and seek advice, green arrows indicate antipathy. A dotted arrow means a weaker link than a solid one. Vertical position in the diagram corresponds to rank. Consider: does the diagram summarize the story about Melissa, Tanya, and Robert in a compact and effective fashion? Suppose you need production to change something so that your stuff will ship sooner — does this diagram let you prioritize who to talk to? Suppose you are an HR manager — does this diagram help you spot effective management strategies?

This example is a toy one — a useful influence map might have dozens of actors and those actors might not be individuals, they might be organizations or even categories of people like “shareholders” or “voters”. Here is a checklist for building an influence map.

  • Figure out which level you are working at — from individuals to broad swaths of society.
  • Come up with a reasonable list of actors — smaller is better, but don’t leave out any important person or group.
  • Decide what types of arrows you are going to use. This depends both on what types of influence you think are present and what kinds of influence you can detect at all. Sometimes there is hidden influence.
  • Decide if some arrow types include others. The black arrow from Melissa to Robert in our example doesn’t mean that Robert doesn’t talk to and seek advice from Melissa — in that diagram black arrows include red arrows.
  • Do some trial sketches so that actors that influence one another are near one another. Any influence map with more than a few people has many different drawings and some have substantially greater clarity than others.
  • Draw! This can be on paper, in a doodling program, or using a formal network management tool.

What can go wrong?

Look at the influence map — with unnamed actors — shown below. The arrow colors have the same meanings as in the Melissa, Tanya, and Robert example. Someone has sketched this diagram while trying to understand why this group of people have trouble making a decision. Here’s a game — based on the diagram, try and explain what the problem is. As with any piece of mathematics, influence maps can be used in a negative way, e.g., to plan sabotage. Occupy Math’s take on this is: please don’t!


Influence maps are a graphical tool for organizing your state of knowledge. They let you figure out who to talk to and who you might need to convince if you want a decision to swing your way. They do not cause you to know more than you know. You can be wrong about the type of influence between two people. A bigger potential problem is that some influence is hidden. Two actors might be neighbors, they might belong to the same service club, or sports team. They might have a past or current relationship that they don’t advertise. The possibilities for hidden influence are many. What can you do about this? If you check the results when you use an influence map against what you thought might happen, it may be possible to detect hidden influence. If Person A always defers to Person B for no reason that makes sense to you, that’s still a link in the influence network. You may want to give those links their own very special color or style, e.g. “tentative links are wavy.”

A good influence map is a permanent work in progress. People come and go or get promoted, relationships change, the situation changes, and you learn things as you go along. Occupy Math uses influence maps to manage departmental and university politics — but those influence maps are mental constructs and they change all the time. Digital or paper influence maps are more useful, perhaps, when trying to manage your neighborhood association or town council, or run a broad campaign to get a new law or regulation adopted or repealed. They are also wizard good for writing proposals because they help you understand why your plan is a good one, or let you make it better. Forgetting a critical constituency can sink a plan and influence maps help you avoid that type of error. Like any other tool, influence maps will become more useful the more practice you get using them.

No, it is too much, let me sum up.

At this point Occupy Math wants to come back to an important point. Influence maps, and more generally directed graphs, are remarkably useful tools. They are also real mathematics and, while today’s post is application-focused and uses only basic math, the topic itself is deep. Directed graphs can be used to teach people kinds of math that will help them become world class scientists or analysts and it also prepares citizens to do well in their lives. It seems bizarre that, in deciding what our children will learn, we focus on topics that are primarily useful for physical science and engineering. Most people don’t end up doing that for a living.

Occupy Math hopes that this post has given you ideas about how to take a little more control of your situation. Occupy Math routinely makes directed graphs of things he’s thinking about, so he is certain that if you start doodling directed graphs they can help you get organized. As always, if you think of other situations where this kind of tool is useful, please dive in and comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics


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