Court the broker, not the rolodex

You have four or five people who could open a door this month, and you are about to start with the one who knows the most names. Write two lists first. List A is high degree: many contacts. List B is high betweenness: they sit on a short path between a group you can already reach and a group you cannot. Spend this month’s introductions on B.

Why these three models

The decision is who receives this month’s introduction capital. Each ask spends someone else’s reputation as well as your time. The features that fire are a market you can only see from where you already sit, a sample of “important people” assembled along those paths rather than drawn from any census, and a scarce connector whose willingness you cannot read on first contact.

Three lenses, three error structures. Network robustness is the complex answer: different scores answer different questions, and a person whose removal splits two groups is a cut vertex. The friendship paradox is the random answer: anyone you meet through a link is more connected than a random draw, which is why A fills the room. Starting small is the equilibrium answer: the first introduction is a low-stakes path, then you scale. You do not lead with the only broker.

The price of an introduction as a loan against the connector’s reputation is already the subject of another piece. I folded it into the third section rather than shipping a fourth card. It adds no lever the three do not already force: treat the scarce ask as scarce.

1. Two lists, and only one of them is scarce

Start with the score, because most founders are running the wrong one.

Different centrality algorithms produce significantly different results, and when the answer looks wrong the first check is whether the algorithm matches the question you actually asked.¹ Degree counts relationships. It is a measure of popularity, or of how freely someone sends messages. Betweenness counts the share of shortest paths that pass through a person. It is a measure of bottlenecks, control points, and vulnerabilities.^{1, 2}

Needham and Hodler put the operator consequence in one sentence. Powerful people are not necessarily in management seats. They sit in brokerage positions, and removing them can seriously destabilise the organisation they sit in.²

A still sharper object lives one definition down. A vertex is a cut vertex if deleting it, and the edges that touch it, disconnects the graph.³ Two distinct biconnected pieces of a network meet in at most one vertex. If they meet in exactly one, that vertex is a cut vertex, and the cut vertices of a graph are the frontiers between those pieces.³ Betweenness scores the frontier. A cut vertex is the frontier with no second path.

Translate that before you dress it up. The person who knows every fintech founder in Nairobi, and who speaks on every panel, is A. The person who is the only reliable path from that circle to the upcountry agent principals, or from your buyers to a bank’s trade desk, is B. B’s phone may be quieter. B is the one whose silence splits the map.

The two scores can disagree sharply. In the textbook Florentine marriage network the Medici only edge the next families on raw degree, about three to two. On betweenness they lie on more than half of the shortest paths between other families; the wealthy rival sits at a tenth.⁶ The usable shape is local: the loud rolodex and the broker are allowed to be different people, and when they are, degree will point you at the wrong one.

Then the bound. Centrality is a ranking of potential impact, not a measure of actual impact. You can identify the two highest-scoring people and still watch policy, custom, or a WhatsApp admin reroute the flow around them.⁴ B is a hypothesis to test.

Needham’s Yelp walk-through makes the same split concrete. Ranking hotel reviewers by PageRank produces one list. Asking which Bellagio reviewers might also act as a bridge between different groups produces another.⁵ If you ranked partners by follower count and then wondered why the introductions stayed inside the room, you asked the popularity question of a brokerage problem.

Robustness-topology, the score lens

• Assumes: importance depends on the question asked, and removing a broker can split groups with no second path.
• Fits because: this month’s introductions are a path problem.
• Breaks when: the two groups already share many overlapping paths, so betweenness is not scarce.
• Evidence: grade B+. Score split and cut-vertex definition are textbook; founder transfer is the B+.
• Counteracts: courting the largest visible rolodex.
• May reinforce: treating a sketch ranking as a person.

2. The names you keep hearing are not a census

The second lens explains why A keeps winning the meeting even after you have named the two scores.

If people compare themselves with their friends, most of them will feel relatively under-connected, because the mean number of friends of friends is always greater than the mean number of friends of individuals.⁷ That is arithmetic. A person with fifty links sits at the end of fifty walks you might take. A person with two sits at the end of two. Follow a relationship, an intro, a group-chat admin, a conference dinner, and you land on the well-connected node far more often than a random draw from the same population would.

Your field of view is built that way. The speaker everyone wants a photo with. The operator who runs three WhatsApp groups. The investor whose name arrives unbidden in every “who should I talk to” reply. They are easy to find because many paths already end on them. A walk through the market is guaranteed to produce them.

You are not under-connected. You are spending on the wrong nodes.

The same bias writes your inbound suggestion list. Ask a connector for “someone well connected in payments” and you will receive A, because that is the node the connector can see. Ask for a path into a named group you cannot currently reach, the agent principals in a specific region, the trade desk at a named bank, the operators who already clear LPOs in a sector you are not in, and you have asked a betweenness question. The second ask is harder to answer. That is the point. An easy answer is usually a hub.

The model breaks when you stop sampling along links. A companies register, a mobile-money merchant list, last year’s exhibitor directory used as a frame rather than as a party: those are censuses. Draw from one of those and the friendship paradox is no longer doing the selecting. Most founders never draw from one of those, because the census is boring and the hub will take the call.

Friendship-paradox, the sample lens

• Assumes: anyone found by following a link is more connected than a random draw.
• Fits because: intro suggestions and “who should I meet” answers arrive along links.
• Breaks when: you sample from a census, a register or a ledger.
• Evidence: grade A-. Feld’s inequality is a theorem; the intro-list transfer is the minus.
• Counteracts: treating the visible market as the market.
• May reinforce: writing off every well-connected person, including real brokers.

3. The first introduction is a path, not the prize

The third lens is how you spend B once you have found them.

Where a counterparty cannot be verified at the outset, an escalating sequence of exposures, each small enough to survive and each larger than the last, builds a record that substitutes for the verification that was unavailable.⁸ Patient players run at a reduced level for a stated number of periods, then move to full effort. Early periods are confidence building. Later periods are where the value is.⁸

Applied here, the first introduction is a low-stakes path: a peer in the same cell, a group add, a twenty-minute coffee that can fail without costing the frontier. Then you scale. The schedule has to be visible enough that a good broker can see there is a larger ask coming, otherwise the small period looks like you do not know what you want.

The ruin case is the one founders reach for when they finally name B. They send the only broker the largest possible ask on first contact, because B is scarce and the month is short. If that forward is lukewarm, or never happens, there is no second path into the block. Sridharan’s frontier is now a wall you built yourself.³

Do not lead with the only broker.

A cut-vertex ask is rationed even when it goes well. Every introduction B makes is a withdrawal from a stock they cannot refill on your timetable. B is also running the same test in the other direction. A first ask that is too large tells them you do not understand the cost.

The folded point sits here. An introduction is collateral the connector posts. A connector who spends that collateral forty times a month has an introduction worth nothing, and everyone in the chain knows it. The only-broker ask should look rare from their side as well as from yours.

Starting-small, the spend lens

• Assumes: counterparties differ in patience, privately, and the scale of the ask can rise over time.
• Fits because: you cannot verify a broker at first contact, and the only path is too expensive to burn.
• Breaks when: the broker’s outside option makes a small first ask not worth sitting through.
• Evidence: grade A. Formally established; the early period is diagnostic.
• Counteracts: leading with the one precious introduction because the month is short.
• May reinforce: never escalating, so you never reach the group you came for.

This month’s introductions, cheapest first

• Write the two lists. One hour, names you already know. A is many contacts. B is on a short path between a group you can reach and a group you cannot. If a name is on both, they are still spent as B, because that is the scarce property.
• Name the two groups for every B. Free, and it is the test. If you cannot name both groups in a sentence, they are not B. They are a person you like.
• Flag the cut vertices. For each B, ask whether any other path into that second group exists. If the answer is no, that ask is rationed. It is not this week’s first message.
• Change the question you put to connectors. “Who is well connected” returns A. “Who can get me a conversation inside this named group” returns a path. The second question is the one that spends the month correctly.
• Hold the only-broker ask. Not forever. Until a cheaper path into the same named group has been tried, or until a dated window will close before a second path can be built.

What to do before you send the next ask

Do now, sized at one hour, effect on the next message. Write list A and list B, and next to each B name the two groups. Reversible, free, and dominant across every story about how well connected you already are. The effect arrives the first time you almost text the loud name and look at the other column instead.

Hedge, premium is two conversations, live this month. Book two low-stakes paths into the group you cannot reach: a peer, a group add, a coffee that can fail. If B was always going to take your call you have delayed the crown ask by a few weeks, and that is the entire downside. If those two paths die, you have learned the frontier is real before you spent it.

Defer and trigger, size written now. Do not send the only-broker ask this week. Pre-commit the trigger: two failed cheaper paths into the same named group, or a dated window (a raise, a licence, a buying cycle) that will close before a second path can be built. Write the sentence of that ask now, while nothing is on fire, because an ask composed in the last fortnight of a window is composed as a plea.

Watch the arrivals. The lists change the next message. What they tell you about a broker arrives only after a path has been attempted, which is weeks, and the temptation to skip to the frontier will peak the first time a coffee is dull.

How a month of loud names usually ends

Run the break test before you blame the market. Has a new fund, a new association, or a licence window added edges that did not exist last quarter? If it has, last year’s brokers may no longer sit on the shortest path, and a ranking you wrote in January is describing a different graph.

If nothing rewired, the common month looks like this. The founder spends the introductions on A. The meetings are pleasant, because A already sits in the founder’s own cell. Nothing opens in the group they actually needed. They conclude the market is closed, or that they need a better deck. The constraint was the path.

The directional base rate is stable. Introductions that stay inside one community feel productive and do not cross a frontier. A conference that returns eight new contacts in your existing WhatsApp neighbourhood has produced degree, which is the wrong score for a group you cannot reach.

Credit the meeting against the meeting you would have got anyway. A coffee with the panel speaker you already share three neighbours with does not update the list of groups you cannot reach. Subtract that before you file the month as business development.

What these three cannot see

None of them measures actual influence. Needham says so, and the honest use of the ranking is as a shortlist of people to test, not as a map of power. A betweenness score computed on a graph you guessed from memory is a ranking of your sketch. It will miss the broker you have never heard of, which is most of them.

They also cannot see a broker who does not want the job. Some people sit on a frontier and charge for it. Some sit on it and are tired. Courting B as if brokerage were a compliment is how you get a polite no that you then file as a closed market.

Screening on patience will also drop brokers with good outside options, because sitting through a small first ask is most expensive for the people who do not need you.

And one property none of the three contains: once you start treating B as the asset, B learns they are the asset, and the price of the next introduction rises. Identification itself moves the node.

The one action that survives the ignorance: before you send the next introduction request, write the two groups that person sits between. If you cannot name both, do not spend the ask. Spend the hour on the lists instead.

Who has to move

The founder who is about to send the WhatsApp, because every connector you already have will hand you A if you ask for someone well connected. The cheapest first test costs an hour and no favours: two columns, the groups named, the cut vertices starred. If B is empty, the next month is for finding one path into one named group, not for collecting more names in the group you already sit in.

Sources and notes

1. Mark Needham and Amy E. Hodler, Graph Algorithms: Practical Examples in Apache Spark and Neo4j, O’Reilly Media, 2019, chapter 5, pages 77 to 78. “Different centrality algorithms can produce significantly different results based on what they were created to measure. When you see suboptimal answers, it’s best to check the algorithm you’ve used is aligned to its intended purpose.” Table 5-1 assigns degree to popularity and gregariousness, closeness to fastest dissemination, betweenness to control points, and PageRank to the transitive influence of neighbours’ neighbours.
2. Needham and Hodler, Graph Algorithms, page 94. Betweenness is used “to find bottlenecks, control points, and vulnerabilities.” “Powerful individuals are not necessarily in management positions, but can be found in ‘brokerage positions’ using Betweenness Centrality. Removal of such influencers can seriously destabilize the organization.”
3. Sriraman Sridharan and R. Balakrishnan, Discrete Mathematics: Graph Algorithms, Algebraic Structures, Coding Theory, and Cryptography, CRC Press / Chapman and Hall, 2019, chapter 1.9.3, pages 58 to 60. “A vertex x is a cut vertex or articulation vertex of the graph G if the removal of the vertex x from G results in a disconnected graph.” Two distinct biconnected components intersect in at most one vertex; if they intersect in exactly one vertex, that vertex is a cut vertex; “the different cut vertices of a graph are the ‘frontiers’ between the biconnected components.”
4. Needham and Hodler, Graph Algorithms, page 99. “Keep in mind that centrality measures represent the importance of a node in comparison to other nodes. Centrality is a ranking of the potential impact of nodes, not a measure of actual impact. For example, you might identify the two people with the highest centrality in a network, but perhaps policies or cultural norms are in play that actually shift influence to others.”
5. Needham and Hodler, Graph Algorithms, chapter 7, pages 157 to 159. The Yelp hotel-reviewer walk-through ranks reviewers by PageRank and then, separately, uses betweenness to find Bellagio reviewers who “are not only well connected across the whole Yelp network, but might also act as a bridge between different groups.” The two lists are not the same.
6. Matthew O. Jackson, Social and Economic Networks, Princeton University Press, 2008, chapter 1, the Florentine marriage example after Padgett and Ansell. The Medici edge the next families on raw degree by a ratio of about three to two; their betweenness is 0.522 against 0.255 for Guadagni and 0.103 for Strozzi. Accessible draft carrying the same calculation: https://web.stanford.edu/~jacksonm/netbook.pdf.
7. Scott L. Feld, Why Your Friends Have More Friends Than You Do, American Journal of Sociology 96(6), 1991, pages 1464 to 1477. The abstract states that the mean number of friends of friends is always greater than the mean number of friends of individuals, and that the disproportion is a class-size paradox rather than a fact about anyone’s character. Open scan: https://www.uvm.edu/pdodds/files/papers/others/1991/feld1991a.pdf.
8. George J. Mailath and Larry Samuelson, Repeated Games and Reputations: Long-Run Relationships, Oxford University Press, 2006. Starting small is section 5.2.4, following Watson: impatient types shirk immediately even against grim trigger; a high enough chance of the impatient type produces perpetual shirking; the separating structure has patient players choose moderate effort for a stated number of periods before escalating, with early periods described as confidence building.

A note on a label this article does not use. Needham describes some large graphs with a hub-and-spoke vocabulary that the complexity literature often tags as scale-free. That label is not imported here, and no exponent is quoted for a customer, referral or introduction network. The usable facts are the score split, the cut vertex, and the sampling bias. They do not require a claim about the tail.

Joshua Agonya Pi’Rwot, Founder.