# Fake Transactions

Minimizing the potential for abuse through fake transactions in the Local network

A service provider may want to artificially increase their graph value, $G_u$ by making fake transactions, which would entitle them to higher rewards. Here, we describe how we minimize this potential abuse vector.

The effect of fake transactions is minimized if the graph value is simply $G_u=\sum_v w(u,v)$. This is because the producer's ownership is directly related only to the total amount of fees that they bring into the network. If the rewards redistributed are based only on the total number of fees, then the producers are on average not going to win anything by performing fake transactions.

Let's instead assume a more generic graph value $G_u$. If the producer adds a set of fake transactions, $f$, that add graph value $G_u^f$ to the producer. Faking these transactions adds a cost $C(G_u^f)$ to the producer, mainly through the network fees they still have to pay.

#### Recovery time

We define $r_u^f$ as the time it takes the user to recover the cost of their fake transactions through rewards.

where we assume the reward period $\tau$ is short enough that we can represent rewards as a continuous integral.

**Things that make fake transactions less profitable:**

- Increasing cost $C(G_u^f)$. In the simple case $G_u=\sum_v w(u,v)$, this is achieved by requiring fee payments to increase your gas value. Any deviation from this metric should also be expensive. If graph connectivity is taken into account, this will become a metric of graph connectivity which will be very expensive to achieve.
- Controlling inflation rate. A slower $M_t$ means performing a set of fake transactions, doesn't immediately translate into owning the corresponding fraction of the supply, since previous token holders still have to be rewarded. Slower inflation can be a tool against fake transactions.
- Redistributing less than the total fees collected. If instead of distributing $R_t$, a smaller fraction $aR_t$, with $a < 1$ is given out, then recovery time becomes longer. Note that we are also free to choose $a > 1$ but this increases the profitability of fake transactions.

### Graph value and fake transactions

One desirable property of the graph value $G_u$, is that it should discourage fake transactions, where one provider $u$ pretends to be a number of different providers and buyers.

Suppose for instance a provider $u$, can instead pretend to be two different fake providers $u_1$ and $u_2$, with a set of fake buyers $V_F$.

Let's assume for now the set of buyers $V_F$ do not earn any rewards (We'll explore deviating from this in the next section), then the condition to prevent fake transactions is that,

That is, the provider will gain nothing by pretending to be two separate providers, and might even benefit more from being one larger provider.

In the case where the graph value is simply the sum of all fees collected, $G_u=\sum_v w(u,v)$, the inequality is automatically satisfied as the two sides are equal, unless $u_1$ and $u_2$ collect more fees than $u$.

Suppose the graph value $G_u$ also incorporates something like the eigenvector centrality. How do we prevent providers from cheating via fake transactions?

Let's define first the normalized eigenvector centrality as a ranking system (similar to Google PageRank). Suppose $A$ is the adjacency matrix of the entire graph, and we calculate the eigenvector, $x$, by solving $\lambda x=A x$ for the largest eigenvalue $\lambda$.

We can produce a producer ranking by ordering the components of $x$, and normalizing $\bar x=x/x_{u_{max}}$, where $x_{u_{max}}$ is the eigenvector centrality of the highest producer. Every producer has a ranking $\bar{x}_u$ with $0\le \bar{x}_u\le 1$.

Now suppose the graph value of a given node is proportional to some function of their eigenvector centrality ranking, $G_u \sim f(\bar x_u)$

We can control how much we prevent fake transactions, by controlling the steepness of the function $f(x)$. Let's consider $u$ splitting into fake producers $u_1$ and $u_2$, to avoid fake transactions, we would need, $f(\bar x_u) \ge f(\bar x_{u_1}) + f(\bar x_{u_2})$ where we can assume $\bar u > \bar{u_1}$ and $\bar u > \bar{u_2}$.

**Given this assumption, we can always choose a function** $f(x)$ **steep enough such that the inequality is satisfied**.

We can define for instance $f(x)=x^\alpha$, and the parameter $\alpha$ can be increased the more we want to prevent producer $u$ from breaking into fake producers $u_1$ and $u_2$.

Note that a trade-off of cracking down on fake transactions this way, is that it ends up rewarding larger, better connected producers, disproportionately more than small producers, so there is a risk that this introduces inequality.

The effective alpha parameter in the Local network plays a crucial role in shaping market dynamics and incentivizing desired behaviors. In mature markets, a high alpha value can lead to excessive concentration of power among established leaders, making it difficult for new entrants to compete. Conversely, in immature markets, a high alpha encourages healthy competition and incentivizes producers to engage in real transactions with users to improve their graph connectivity.

We can adjust the alpha parameter based on market maturity. When launching a new market, a high alpha value promotes competition and helps bootstrap the market by offering higher rewards. As the market matures and dominant producers emerge, lowering the alpha value can prevent excessive concentration of power and ensure a more balanced ecosystem.

### Temporal Considerations of Effective Alpha

In a mature market, there is already an established set of providers, and it is difficult for a new producer to join the market and climb to the highest rank. This means that for a given producer $u$, it is already difficult to increase their graph value $G_u$ by increasing their eigenvector centrality ranking. This effectively behaves similarly to an immature market with a higher value of $\alpha$.

The parameter $\alpha$ can be adjusted by governance based on the maturity of the market. A large $\alpha$ parameter in a mature market can have a negative effect. In a mature market there are already established leaders that are hard to overtake, and a large $\alpha$ then disproportionately rewards these leaders even more and makes it even harder to overtake.

In an immature market, there are no established leaders, and there can be free active competition to try to become the leader. This competition is welcome and beneficial to reach market maturity. Introducing a large $\alpha$ parameter here would increase the incentive to compete to dominate the market. At this stage any producer that dominates could still be overtaken by a competitor, so the risk of excessively concentrating power is not as large as in a mature market.

**Driving good competition with $\alpha$:** In an immature market the risk of a producer increasing their relative through fake transactions is higher, since it is easier to rise in the provider ranking. A large $\alpha$ means that the best strategy to increase your power as a producer, is to have a more connected vertex in the graph.

Connectivity is difficult to manufacture through fake transactions. If one producer sets up a set of fake buyers that they will transact with, this graph of the producer with its fake buyers will still be disconnected from the rest of the graph. The only way to increase connectivity is to generate transactions from real buyers who themselves also complete transactions with other producers, who themselves complete transactions with other buyers...

Our proposal is to keep $\alpha$ large when a new market is launched, which will encourage competition to become the dominant producer, by completing real transactions with real users which will improve their graph connectivity. A high $\alpha$ also enables us to bootstrap the emerging market by offering higher rewards than the total fees collected. As the market matures, and dominant producers are established, $\alpha$ can be lowered to avoid excessive concentration of power, together the lowering of rewards.