Graph generation model

We assume discrete time and that no two users arrive simultaneously at the poll, i.e. one user arrives at time

$t$ and the other at time

$t^{{}^{\prime }}$, such that

$t<t^{{}^{\prime }}$.

At time

$t=0$, we have

$\left|U_0\right|=0,\left|V_0\right|=n_0^v,\left|E_0\right|=0$. This reflects the fact that the poll starts with no users, a set of initial statements and no votes.

At time

$t-1$, we consider the graph

$G_{t-1}=(U_{t-1},V_{t-1},E_{t-1})$ as described in the previous section.

At each time

$t$, either a new user

$u_t\notin U_{t-1}$ participates in the poll (such that

$u_t\cup U_{t-1}=U_t$) or an existing user

$u\in U_{t-1}$ returns.

• A new user
  $u_t$ votes for
  $0$ or
  $k\ge 1$ existing statements
  $v_t^1,\dots ,v_t^k\in V_{t-1}$. Voting creates the corresponding edges,
  $(u_t,v_t^1),\dots ,(u_t,v_t^k)$, such that
  $E_{t-1}\cup \left\{(u_t,v_t^1),\dots ,(u_t,v_t^k)\right\}=E_t$, or no edges in case of no voting (implying
  $E_{t-1}=E_t$). User
  $u_t$ can also create
  $0$ or
  $l\ge 1$ new statements
  $v_t^1,\dots ,v_t^l\notin V_{t-1}$, such that
  $V_{t-1}\cup \left\{v_t^1,\dots ,v_t^l\right\}=V_t$ (if no statements are created,
  $V_{t-1}=V_t$).
• An existing, returning user
  $u$ visited the poll latest at time
  $t^{{}^{\prime }}<t$, either as a new user or returning one (i.e. an existing user may return multiple times). On time
  $t$ they vote for
  $0$ or
  $k\ge 1$ existing statements
  $v_t^1,\dots ,v_t^k\in V_{(t^{{}^{\prime }},t)}$, where
  $V_{(t^{{}^{\prime }},t-1)}$ is the set of all statements added to the poll between time
  $t^{{}^{\prime }}$ and
  $t-1$ (possibly $\emptyset $ if
  $t^{{}^{\prime }}=t-1$). Similarly to the previous case, user
  $u$ can also create
  $0$ or
  $l\ge 1$ new statements
  $v_t^1,\dots ,v_t^l\notin V_{t-1}$, such that
  $V_{t-1}\cup \left\{v_t^1,\dots ,v_t^l\right\}=V_t$ (if no statements are created,
  $V_{t-1}=V_t$).

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