Remark on the value of $\varepsilon$. A suitable value for $\varepsilon$ can be $\varepsilon = \Theta(n^{-1/4})$ for which we get a runtime of $\til O(\mathcal T \sqrt n + kn)$, a success probability of $1 - O(\log n / \sqrt n)$ and an approximation guarantee of $1 - \gamma_f - \Theta (n^{-1/4})$.

* Adaptation of the Rounding Scheme of [58] for general bailouts. We can adapt the dependent rounding scheme of [58] to perform randomized rounding such that (i) we obtain the same approximations as in the case of independent rounding; (ii) the constraints of the problem are satisfied with high probability. The problem that the independent rounding scheme has is that there exist choices of the entries of $L$ (e.g. $L_j = 2^{j-1}$) such that $L^T Z$ is anti-concentrated. In fact, when $L_j = 2^{j - 1}$ and $Z \sim \mathrm{Be} (1/2 \cdot \one)$ then $L^T Z \sim \mathrm{Uniform}(\{ 0, \dots, 2^n - 1 \})$. To mitigate this problem we will use the main result of  [58] as follows: We first solve the LP relaxation and get the optimal fractional bailouts $\til z^*$ as in the independent rounding case.  Then we define a vector $U$ as a sample from the oracle of [58] (described in Section 2, and Section 3.2 that modifies the original algorithm for a non-integral sum)

\begin{equation*}
\small
U \sim \D_{\mathrm{dep}} \left ( \underbrace {\frac {L_1} {\| L \|_{\infty}} (1 - \til z_1^*), \dots, \frac {L_n} {\| L \|_{\infty}} (1 - \til z_n^*), \left \lceil \frac {\| L \|_1 - \Lambda} {\| L \|_{\infty}} \right \rceil -  \frac {\| L \|_1 - \Lambda}  {\| L \|_{\infty}}}_{\text{probabilities}}; \underbrace {n + 1}_{\text{\# variables}} ; \underbrace{\left \lceil \frac {\| L \|_1 - \Lambda} {\| L \|_{\infty}} \right \rceil}_{\text{total weight}} \right ),
\end{equation*}

and let $Z = \one - U$. For (i) we have that $\ev {} {Z_j} = \Pr [ Z_j = 1] = 1 - \Pr [ U_j = 1] = 1 - \tfrac {L_j} {\| L \|_{\infty}} (1 - \til z_j^*) \ge \til z_j^*$ and thus we can achieve the desired approximation guarantees of <ref>. For (ii) we know that $\sum_{j} L_j U_j$ are concentrated and thus we can easily show that $\sum_{j} L_j Z_j$ are concentrated as well, and thus with high probability the rounded bailouts are at most $\Lambda + \til O(\sqrt \Lambda)$. Finally, the runtime can be similarly deduced to be $\til O \left ( \tfrac {\mathcal T} {\varepsilon^2} + \tfrac {\Lambda} {L_{\min} \varepsilon^4} \right )$ for a success probability of $1 - O \left ( \tfrac {\log n} {n \varepsilon^2} \right )$. Thus, again, for $\varepsilon = \Theta (n^{-1/4})$ we get an approximation ratio of $1 - \gamma_f - \Theta (n^{-1/4})$ with probability $1 - O \left ( \tfrac {\log n} {\sqrt n} \right )$.

Proof of <ref>.
We start with the complete network on $n$ nodes $K_n$, with $c_i = b_i = 1$ and $p_{ij} = 1$, and a point-mass shock of $X_i = \varepsilon$ for some $\varepsilon \in (0, 1)$. We also let $L = n \varepsilon / k \cdot \one$ and $v > \zero$. In the fractional optimum  we have that $\til z_i^* = k / n$ and the optimum value of $\| v \|_1 \cdot  n$ is achieved. When we choose to round to $k$ nodes, we do that uniformly (since $\til z_i^* =  k / n$) over all $k$-sets, that is the probability of each $k$-set being selected is $\tfrac 1 {\binom n k}$. Thus the value optimum is just the value of the optimum given that the values of any $k$-set are set to 1 (w.l.o.g we can choose the $k$-set $\{ 1, \dots, k \}$). The rounded nodes' bailouts would suffice to avert the shock, so they will be truncated to have a value of $\varepsilon$, from the problem constraints (debts must only be paid in at most their full value). Since all nodes are marginally default (that holds for the bailed-out nodes with the truncated values as well), we have from
<ref> (with equality since every node satisfies the constraint with tight equality) that $(1 / n) \sum_{j \in [n]} \bar p_j = n - n \varepsilon + k \varepsilon$
which implies that $\opt_{f} = v^T \bar p \le \| v \|_\infty \left [ n (n - \varepsilon (n - k)) \right ]$.
We thus have $\sigma_f \ge \tfrac {\zeta^{-1}} {1 - \varepsilon (1 - k / n)}$.
So for $k = o(n)$ (small) and $n$ large the gap tends to $\tfrac {\zeta^{-1}} {1 - \varepsilon}$ which can be arbitrarily bad for $\varepsilon \to 1$.

Proof of <ref> (see <ref>). Let $a (| \mathcal I|)$ be a poly-time computable function on the input size $| \mathcal I |$, and $\alpha \in (0, 3)$. We construct the same proof as the hardness reduction of  <ref> with the only change that instead of adding one copy of “element nodes” we add $a (| \mathcal I |)$ copies of “element nodes” resulting in a network with a polynomial number, i.e. $m + a(| \mathcal I | ) \cdot n$, of nodes. We connect, with direction from left to right,  the element nodes with liabilities $\tfrac {1 - \alpha / 3} n$ (i.e. a fully-connected network) and we set the external liabilities of the $a(| \mathcal I|)$-th level to be $1 - \alpha / 3$. We distinguish the following cases: If the answer to the 3-Set-Cover problem is YES then there are at least $k + a(| \mathcal I|) \cdot n$ solvent nodes. Else, if the answer to the 3-Set-Cover problem is NO then there are at most $k + n$ solvent nodes. That is, if we were able to distinguish between the (<ref>) being between $k + n$ and $k + a(| \mathcal I|) \cdot n$ in polynomial time we will be able to solve the 3-Set-Cover in polynomial time, which is a contradiction assuming that P $\neq$ NP. The approximation gap is $\tfrac {k + n \cdot a(| \mathcal I |)} {k + n} \ge \tfrac {a(| \mathcal I |) \cdot n } {2n} = \tfrac {a(|\mathcal I|)} {2} = \Omega \left ( a (| \mathcal I |) \right )$.

Proof of <ref>.

* GC Problem. We omit the dependence on $g$ to lighten notation. We have that

\begin{align*}
\evz {\sum_{j \in [n]} Z_j} & \ge \frac 1 {2ng} \evz { \sum_{i, j \in [n]} |Z_i - Z_j| } \explain {Gini Constraint} \\
& = \frac 1 {2ng} \sum_{i, j \in [n]} \left ( \til z_i^* (1 - \til z_j^*) + \til z_j^* (1 - \til z_i^*) \right ) \explain{Definition of Expected Value} \\
& \ge \frac 1 {ng} \left ( n \sum_{j \in [n]} \til z_j^*  - \sum_{i, j \in [n]} \til z_i^* \til z_j^* \right ) \explain{Reorder Summation} \\
& \ge \frac 1 {ng} \left ( n \sum_{j \in [n]} \til z_j^*  - k \sum_{j \in [n]} \til z_j^* \right ) \explain{At most $k$ nodes are bailed out} \\
& = \frac 1 g \left ( 1 - \frac k n \right ) \sum_{j \in [n]} \til z_j^*
\end{align*}

If $\tfrac 1 g \left ( 1 - \tfrac k n \right ) < 1$, or $g > \left ( 1 - \tfrac k n \right )$ then we follow the same analysis as in <ref> and obtain an $\tfrac {(1 - \beta_{\max}) (1 - k  / n)} g$ approximation guarantee. We can use concentration inequalities to devise bounds about the runtime of the algorithm in a similar manner to <ref>.

Proof of <ref>. General Instance. Suppose that we have the star network $S_n$ with node 1 being in the center and nodes 2 through $n - 1$ being the peripheral nodes. Let $b = \one$, $c = n \cdot \one_{\{ 1 \}}$, and $p_{1i} = 1$ for $i \neq 1$. The bailouts are $L = n \cdot \one$ and the total budget is $\Lambda = n$.
The network is hit with a point-mass shock $X = c$. We will (w.l.o.g.) focus on the (<ref>) objective [It can be shown for any linear objective given by a vector of coefficients $v > \zero$ the same result holds].
The unbounded objective has value $2n - 1$ since bailing out the central node restores the network to its initial state.  Let $g = 0$. We have the following results:

* (<ref>) Solution. Subject to absolute equality (i.e. $g = 0$), it should hold that for the bailout indicator variables $\bar z_j$ that $\sum_{j \neq 1} | \bar z_1(g) - \bar z_j(g) | = 0$. Thus $\bar z_j(g) = 0$, therefore the objective value is $0$. That yields a PoF of $\tfrac {2n - 1} {0} = \infty$.

* (<ref>) Solution. Let $q = \one_{\{ 1 \}}$. Again, similarly to the previous case, we get $\bar z_j(g, q) = 0$ for all $j$. Thus the PoF is  $\infty$.

* (<ref>) Solution. We again get that $\sum_{(i, j) \in E} a_{ij} |\bar z_i(g, A) - \bar z_j(g, A)| = 0$ which yields $\bar z_j(g, A) = 0$ for all $j \in [n]$. Again, the PoF is $\infty$.

Proof of <ref>. We will prove the correctness of the Theorem for the (<ref>) metric since the other metrics have exactly the same proof. Fix some $g \ge 0$. The fractional PoF is equal to $f(\til p^*(1)) / f(\til p^*(g))$. Assume, for contradiction, that the PoF is unbounded. The following can happen:

* $f(\til p^*(g)) \neq 0$ and $f(\til p^*(1))$ is such that the limit of their ratio goes to infinity. This is a contradiction since $f$ can get values up to $\| f \|_\infty$. Thus, we arrive at a contradiction.

* $f(\til p^*(g)) = 0$. Since $f$ is strictly increasing with $f(\zero) = 0$ then $\til p^*(g) = \zero$. We will show that $\til p^*(g) = \zero$ if and only if $X = c$ w.p. 1 and $\til z^*(g) = 0$. We remind here that there are no isolated nodes (to the internal or the external sector) and hence $p > \zero$. The reverse direction is trivial since the fairness constraint is always satisfied (hence it does not affect the feasible region) and the fixed point operator is simply $\Phi(\til p^*(g)) = p \wedge (A^T \til p(g))$, so $\Phi(\zero) = p \wedge \zero = \zero$, i.e. $\til p^*(g) = \zero$. For the reverse direction, since $p > \zero$ the only way for $\til p^*(g)$ to be $\zero$ is (i) $g = \Lambda = 0$ which is impossible since by the definition of the problem $\Lambda > 0$, and (ii) the relative liability non-homogeneous part is zero, i.e. $(c - X) + L \odot\til z^*(g) = 0$ (the fairness constraint is trivially satisfied). Since $c - X \ge \zero$, $L > \zero$ and $\til z^* \ge 0$, the only way for the equation to hold is $X = c$ with probability 1 and $\til z^*(g) = \zero$, yielding a contradiction. Therefore, $f(\til p^*(g)) > 0$.

Proof of <ref>.

* (<ref>) Solution. Since $g > 0$ and $A$ is a connected network (i.e. no isolated nodes) a positive budget $\mu \in (0, \Lambda]$ is spent in total on bailouts. We let
$$\B(\mu) = \{ (\til p'(g, A), \til z'(g, A))^T \text{ feasible $(g, A)$-unfair allocation} : L^T \til z' = \mu \}.$$

Subsequently, we define the following partition of $\B (\mu)$: Let $\B_1(\mu) = \B (\mu) \cap \{ \exists j \in [n]:  \til z_j(g, A) \neq \mu /  (n L_j) \}$, and $\B_2(\mu) = \B (\mu) \cap \{ \forall j \in [n] : \til z_j(g, A) = \mu /  (n L_j) \}$.
We show that if $\B_2(\mu)$ is non-empty then $\B_1(\mu)$ must be non-empty. Let $\til z(g, A) \in \B_2 (\mu)$. Then since $g > 0$ for $0 < \delta_{ij} \le \min \{ \tfrac {\mu} {nL_i}, \tfrac {\mu} {nL_j}, \tfrac {g \beta_{\min} \mu} {\beta_i + \beta_j} \}$ we choose two arbitrary nodes $i$ and $j$ such that the former node's bailout is increased by $\delta_{ij}$ and the latter node's bailout is decreased by $\delta_{ij}$, that is absolute equality is achieved via the optimal allocation and we are allowed to increase up to $g$. The total budget constraint is not violated and the new bailout scheme belongs to $\B_1(\mu)$.  If the optimal bailout scheme belongs to $\B_2(\mu)$ perturb the bailouts of two nodes as we described above to get a sub-optimal assignment $(\til p'(g, A), \til z'(g, A))^T \in \B_1(\mu)$ since the optimal assignment is also $(0, A)$-unfair. The altered assignment is a lower bound on the original assignment per the objective. Below we bound the PoF for the (<ref>) objective. We have:

\begin{align*}
\frac 1 {\mathrm {PoF}_{(g, A)}} &= \frac {\sum_{j \in [n]} v_j \til p_j^*(g, A)} {\sum_{j \in [n]} v_j \til p_j^*(1, A)} \\
& \ge \frac 1 \zeta \cdot \frac {\sum_{j \in [n]} \til p_j'(g, A)} { \sum_{j \in [n]} \til p_j^*(1, A)} \explain{Constrain to $\B_1(\mu)$} \\
& = \frac 1 \zeta \cdot \frac {\sum_{j \in D(g, A)} \til p_j'(g, A) + \sum_{j \in R(g, A)} \til p_j'(g, A)} {\sum_{j \in D(1, A)} \til p_j^*(1, A) + \sum_{j \in R(1, A)} \til p_j^*(1, A)} \explain{Partition of $[n]$} \\
& \ge \frac 1 \zeta \cdot \frac {\sum_{j \in D(g, A)} \til p_j'(g, A)} {\sum_{j \in D(1, A)} \til p_j^*(1, A) + \sum_{j \in R(1, A)} \til p_j^*(1, A)} \explain{Drop Summands} \\
& \ge \frac {1 - \beta_{\max}} {\zeta} \cdot \frac {\sum_{j \in D(g, A)} (c_j - x_j + L_j \til z_j'(g, A))} {\sum_{j \in [n]} (c_j - x_j + L_j \til z_j^*(1 , A))} \explain{\cref{lemma:lower_bound,lemma:bounding}} \\
& \ge \frac{1 - \beta_{\max}} {\zeta} \cdot \frac {\sum_{j \in D(g, A)} (c_j - x_j + \beta_j L_j \til z_j'(g, A))} {\sum_{j \in [n]} (c_j - x_j) + \Lambda} \explain{Budget Constraint, $\beta_j < 1$} \\
& \ge \frac {1 - \beta_{\max}} {\zeta} \cdot \frac {\sum_{j \in D(g, A)} (c_j - x_j) + \frac 1 {2g} \sum_{(i, j) \in E} a_{ij} |L_i  \til z_i'(g, A) - L_j \til z_j'(g, A)|} {\sum_{j \in [n]} (c_j - x_j) + \Lambda} \explain{Gini Constraint} \\
& \ge \frac {1 - \beta_{\max}} {2g \zeta} \cdot \frac {\sum_{(i, j) \in E} a_{ij} |L_i  \til z_i'(g, A) - L_j \til z_j'(g, A)|} {\| c \|_1 + \Lambda} \explain{$\zero \le c - x \le c$} \\
& \ge \frac {1 - \beta_{\max}} {2g \zeta (\| c \|_1 + \Lambda)} \sqrt {\sum_{(i, j) \in E} a_{ij}^2 (L_i  \til z_i'(g, A) - L_j \til z_j'(g, A))^2} \explain{$\| x \|_1 \ge \| x \|_2$} \\
& \ge \frac {1 - \beta_{\max}} {2g \zeta (\| c \|_1 + \Lambda)} \sqrt {\til \theta'(g, A)^T \L(A \odot A) \til \theta'(g, A)} \explain{$\til \theta'(g, A) = L \odot \til z'(g, A)$}. \\
\end{align*}

We do a change of variables $\til y'(g, A) = \til \theta'(g, A) - \tfrac \mu n \one \neq \zero$ since $\theta'(g, A) \in \B_1(\mu)$. Since $\one$ is the eigenvector of the 0 eigenvalue we can center the aforementioned quadratic form to be equal to

$$\til y'(g, A)^T \L (A^{(2)}) \til y'(g, A) = \frac {\til y'(g, A)^T \L (A^{(2)}) \til y'(g, A)} {\| \til y'(g, A) \|_2^2} \cdot \| \til y'(g, A) \|_2^2.$$

$$ \frac {\til y'(g, A)^T \L (A^{(2)}) \til y'(g, A)} {\| y'(g, A) \|_2^2} \cdot \| \til y'(g, A) \|_2^2 \ge \left ( \min_{\one^T y(g, A) = 0}  \frac { y(g, A)^T \L (A^{(2)}) y(g, A)} {\| y(g, A) \|_2^2} \right ) \cdot \| \til y'(g, A) \|_2^2.$$

Using the Courant-Fischer Theorem [57] the above expression evaluates to $\lambda_2(\L (A^{(2)})) \cdot \| \til y'(g, A) \|_2^2$, where

$$\| \til y'(g, A) \|_2^2 = \sum_{j \in [n]} \left ( L_j \til z_j' \right )^2 - \frac {\mu^2} n > 0.$$

For $\lambda_2( \L (A^{(2)}) )$ we use the lower bound provided by Cheeger's inequality, i.e.

\begin{equation*}
\lambda_2 ( \L (A^{(2)}) ) \ge \frac {\phi^2(A^{(2)})} {2 \max_{j \in [n]} \sum_{i \in [n]} a_{ji}^2} \ge \frac {\phi^2(A^{(2)})} {2 \max_{j \in [n]} \left ( \sum_{i \in [n]} a_{ji} \right )^2} \ge \frac {\phi^2(A^{(2)})} {2 \beta_{\max}^2},
\end{equation*}

where $\phi(A^{(2)}) = \min_{\emptyset \subset S \subset [n]} \frac {\sum_{j \in S, i \notin S} a_{ji}^2} {\min \left \{ \sum_{j \in S} \sum_{i : i \sim j} a_{ji}^2 , \sum_{j \in [n] \setminus S} \sum_{i : i \sim j} a_{ji}^2  \right \}}$ is the conductance of $A^{(2)}$. Thus we can bound the PoF by

\begin{equation*}
\mathrm{PoF}_{(g, A)} \le \frac {\sqrt 2 g \zeta (\| c \|_1 + \Lambda) \beta_{\max}} {(1 - \beta_{\max}) \phi(A^{(2)}) \cdot \sqrt {\sum_{j \in [n]} (L_j \til z_j')^2 - \mu^2 / n}}.
\end{equation*}

Note that the denominator is always non-zero since $\beta_{\max} < 1$, $\phi(A^{(2)}) > 0$ and $\sum_{j \in [n]} (L_j \til z_j')^2 - \mu^2 / n > 0$ due to $\til z'(g, A) \in \B_1(\mu)$.

* (<ref>) Solution. The construction of the sets $\B(\mu), \B_1(\mu), \B_2(\mu)$ is exactly the same as in the previous case, as well as the derivation of the PoF bound. The only change that in place of $A^{(2)}$ we put the complete graph $K_n$ on $[n]$. Since $\lambda_2 (\L (K_n)) = n$ we do not need to employ Cheeger's inequality, and thus arrive at a bound of the form

\begin{equation*}
\mathrm{PoF}_{g} \le \frac {2 g \zeta (\| c \|_1 + \Lambda) \sqrt n} {(1 - \beta_{\max}) \sqrt {\sum_{j \in [n]} (L_j \til z_j')^2 - \mu^2 / n}}.
\end{equation*}

Note that there was a factor of $2n$ on the Gini Coefficient definition so the bound gets a $2 \sqrt n$ factor on the numerator. Again for a finite instance we have that $\mathrm{PoF}_g < \infty$.

Proof of <ref>. We have that

\begin{align*}
\mathrm{SGC}(\til z; A) = \frac {\sum_{i, j} a_{ij} |L_i \til z_i(g, A) - L_j \til z_j(g, A)|} {2 \sum_{j \in [n]} \beta_j L_j \til z_j(g, A)} = \frac {1} {2} \psi(L \odot z; A).
\end{align*}

where $\psi(x; A) = \frac {\sum_{i, j} a_{ij} | x_i - x_j |} {\sum_{j \in [n]} \beta_j |x_j|} $. In <ref>, we prove that

\begin{equation*}
\phi(A) = \min_{x \neq \zero, \; \mathrm{median} (x) = 0} \psi(x; A).
\end{equation*}

Thus $\tfrac {\psi(L \odot z; A)} {2} \ge \tfrac {\phi(A)} {2}$. Therefore $\mathrm{SGC}(\til z; A) \ge \tfrac {\phi(A)} {2}$.

§ TRANSFORMATIONS

§.§ Transforming Weakly-Increasing Objectives to allow clearing vectors

In the case that the (<ref>) problem consists of a function $f(\bar p)$ which is increasing (i.e. not necessarily strictly increasing), there is no guarantee that the solution that will be produced by the optimization procedure will correspond to a clearing vector (still if the spectral radius of $A$ is less than 1 the fixed point operator yields a unique solution). For that reason we consider, similarly to [3] the transformed objective, parametrized by $\epsilon > 0$,

\begin{equation*}
\hat f(\bar p) = f(\bar p) + \frac {\epsilon (1 - \beta_{\max})} {2 \Lambda} \one^T \bar p.
\end{equation*}

It is obvious that for any $\bar p > 0$ we have that $\hat f(\bar p) > f(\bar p)$ since the added term contributes positively. Moreover $\hat f$ is strictly increasing and therefore an optimal solution $\hat p$ to $\hat f$ subject to the (<ref>) constraints is a clearing payment vector. Below we will show that if $\bar p^*$ is an optimal solution to $f$, and $\hat p^*$ is an optimal solution to $\hat f$ then $f(\hat p^*) - f(\bar p^*) \le \epsilon$. First, from optimality we have that

\begin{equation*}
\hat f(\hat p^*) - \hat f(\bar p^*) \ge 0.
\end{equation*}

\begin{equation*}
f(\hat p^*) + \frac {\epsilon (1 - \beta_{\max})} {2 \Lambda} \one^T \hat p^* - f(\bar p^*) - \frac {\epsilon (1 - \beta_{\max})} {2 \Lambda} \one^T \bar p^* \ge 0.
\end{equation*}

\begin{equation*}
f(\hat p^*) - f(\bar p^*) \le \frac {\epsilon (1 - \beta_{\max})} {2 \Lambda} \| \hat p^* - \bar p^* \|_1 \le \frac {\epsilon (1 - \beta_{\max})} {2 \Lambda (1 - \beta_{\max})} \| L^T \hat z^* - L^T \bar z^* \|_1 \le \epsilon.
\end{equation*}

Where the inequalities follow from rearrangements, the triangle inequality, <ref>, and that the maximum distance between any two assignments can be at most $2 \Lambda$. Thus the clearing payment vector from solving the transformed objective yields an objective value which is $\epsilon$-close to the desired optimal value.

§.§ A Property of Conductance

To conclude the proof of <ref> we give a relation between the conductance and Gini-related Measures. For this subsection only, and for simplicity, we will assume a new notation from the rest of the paper referring to an unweighted and undirected topology $G(V = [n], E)$ with $|V| = n$. The result we prove below is also of independent interest from the rest of the paper. The conductance of $G$, $\phi(G)$, is given as

\begin{equation*}
\phi(G) = \min_{\emptyset \subset S \subset V} \frac {e(S, \bar S)} {\min \{ |S|, | \bar S | \}} =  \min_{\emptyset \subset S \subset V, |S| \le n / 2} \frac {e(S, \bar S)} {|S|},
\end{equation*}

where $e(S, \bar S)$ is the number of edges going from $S$ to its complement  $\bar S = [n] \setminus S$. More specifically, we will show that

\begin{align*} \tag{COND} \label{eq:cond}
\phi(G) &  = \min_{x_1, \dots, x_n} & \frac {\sum_{(i, j) \in E} |x_i - x_j|} {\sum_{i \in [n]} |x_i|} \\
& \text{subject to} & \mathrm{median}(x_1, \dots, x_n) = 0 \\
& & \text{not all $x_i = 0$}.
\end{align*}

For that reason, we define the function

\begin{equation*}
\psi(x) = \frac {\sum_{(i, j) \in E} |x_i - x_j| } {\sum_{i \in [n]} |x_i|}.
\end{equation*}

Firstly, we will show that $\min_{x \in \eqref{eq:cond}} \psi(x) \le \phi(G)$. Let $S^*$ be the set that achieves $\phi(G)$. We let

\begin{equation*}
x_i^* = \begin{cases}
\frac {1} {\phi(G)} & i \in S^* \\
0 & i \notin S^*
\end{cases}.
\end{equation*}

Note that since $|S^*| \le n / 2$ the median of $x_i^*$ is 0. Moreover, not all $x_i^*$ are 0. We have that

\begin{equation*}
\psi(x^*) = \frac {\sum_{(i, j) \in E} | x_i^* - x_j^* | } {\sum_{i \in [n]} |x_i^*|} = \frac {\sum_{(i, j) \in E, i \in S^*, j \in \bar S^*} \frac {1} {\phi(G)}} {|S^*| \frac {1} {\phi(G)}} = \frac {e(S^*, \bar S^*)} {|S^*|} = \phi(G).
\end{equation*}

Thus we showed that $\min_{x \in \eqref{eq:cond}} \psi(x) \le \phi(G)$. We will now show that $\min_{x \in \eqref{eq:cond}} \psi(x) \ge \phi(G)$. In this case, we seek to find $x_1, \dots x_n$ such that $\psi(x)$ is minimized and the $x_i$'s have median 0. We define $y_i = \max \{ x_i, 0 \}$ and $z_i = \min \{ x_i, 0 \}$. It is elementary to show that $|x_i| = |y_i| + |z_i|$ and $|x_i - x_j| = |y_i - y_j| + |z_i - z_j|$. So

\begin{equation*}
\psi(x) = \frac {\sum_{(i, j) \in E} \left [  |y_i - y_j| + |z_i - z_j| \right ]} {\sum_{i \in [n]} \left [|y_i| + |z_i| \right ]}.
\end{equation*}

To show our claim, it suffices to show that $\psi(y) \ge \phi(G)$ and $\psi(z) \ge \phi(G)$. This is because for every $a, b, c, d, \alpha \ge 0$ with $\tfrac {a} {c} \ge \alpha$ and $\tfrac {b} {d} \ge \alpha$ then $\tfrac {a + b} {c + d} \ge \alpha$. By symmetry, we will constrain ourselves to proving $\psi(y) \ge \phi(G)$, whereas for $z$ the proof is identical. We let $y_1 = \dots = y_{n / 2} = 0$ and $y_i \ge 0$ for all $i \ge n / 2$. This guarantees that the median of $y_i$ is 0 and that $y_i \ge 0$ for all $i \in [n]$. For the numerator we have

[size=0.8, x=-4, y=0]1
[x=-4, y=0, distance=0.5, position=below]${y_1 = \dots = y_{n / 2} = 0}$
[size=0.8, x=-2, y=0, label=${y_i}$]2
[size=0.8, x=-1, y=0, label=${y_{i + 1}}$]3
[size=0.8, x=-0, y=0, label=${y_{i + 2}}$]4
[x=1, y=0]$\dots$
[size=0.8, x=2, y=0, label=${y_j}$]5
[Pseudo, x=0.75, y=1.5]6
[Pseudo, x=0.75, y=-1.5]7
[color=gray,bend=30, style=dashed](6)(7)
[x=0.75,y=-1.5, position=right, distance=0.5,color=gray]$C_\ell, \; \ell - 1 \ge n / 2$

\begin{align*}
\sum_{(i, j) \in E, i < j} (y_j - y_i) & = \sum_{(i, j) \in E, i < j} \sum_{\ell = i + 1}^{j} \left ( y_{\ell} - y_{\ell - 1} \right ) \\ & =  \sum_{\ell} \left ( {\text {\# of copies of the $(\ell - 1, \ell)$ segment}} \right ) \cdot \left ( y_\ell - y_{\ell - 1} \right ) \\ & = \sum_{\ell} \left ( \text{\# of edges that cross over $(\ell-1, \ell)$} \right ) \cdot (y_{\ell} - y_{\ell - 1}) \\
& = \sum_{\ell} \underbrace{\left ( \text {\# edges in the cut $C_\ell$ partitioning $[\ell - 1]$ from $[n] \setminus [\ell - 1]$ }\right )}_{| C_\ell |} \cdot ( y_{\ell} - y_{\ell - 1} )
\end{align*}

A key observation is that $\tfrac {|C_\ell|} {n - \ell + 1} \ge \phi(G)$. Therefore the numerator satisfies

\begin{equation*}
\sum_{\ell} |C_\ell| \cdot (y_{\ell} - y_{\ell - 1}) \ge \sum_{\ell} \phi(G) (n - \ell + 1) (y_{\ell} - y_{\ell - 1}) \ge \phi(G) \cdot \sum_{\ell} y_\ell = \phi(G) \cdot \sum_{\ell} |y_\ell|.
\end{equation*}

Thus $\psi(y) \ge \phi(G)$. We conclude that $\phi(G) = \min_{x \in \eqref{eq:cond}} \psi(x)$. <ref> is the weighted version (with weights $w_{ij} \ge 0$) of this identity where the proof is exactly the same, where we use the function $\psi(x; W) = \frac {\sum_{(i, j) \in E} w_{ij} |x_i - x_j|} {\sum_{i \in [n]} d_i |x_i|}$ and $d_i = \sum_j w_{ij}$. Finally, note that if we constrain $\psi$ in the domain in which the denominator is at most $\sigma'$, for some $\sigma' > 0$ then we modify an allocation $x$ with $\sum_{i \in [n]} |x_i| = \sigma > 0$ to an allocation $x' = \tfrac {\sigma'} {\sigma} x$ and the proof will remain valid since $\psi$ is scale-invariant, that is $\psi(x') = \psi \left ( \tfrac {\sigma'} {\sigma} x \right ) = \psi(x)$.

§ DATASETS ADDENDUM

German Banks Dataset [22]. The dataset contains $n = 22$ German Banks, connected with a liability matrix with $m = 435$ edges, with a mean outdegree $\bar d_{\mathrm{out}} = 19.8$. The network structure can be described by the ER network $G(n = 22, p = 0.94)$. Below we provide Pareto fits for the internal liabiliti-es distributions, a regression plot for external assets and external liabilities, a plot of the financial network with equities (wealths) displayed with colors on a log-scale, and a distribution plot between external assets and external liabilities with Pareto fits. Observe that the relation between external assets and external liabilities is a linear function with $R^2 = 0.999$. The visualization of the dataset appears in  <ref>.

[Internal liabilities distribution and Pareto fit]
[External assets and external liabilities regression]
[Financial network plot with wealths on log-scale]
[External assets and liabilities distributions and Pareto Fits]

German Banks dataset statistics

EBA Dataset [34]. The EBA dataset contains information about the external assets and liabilities of each bank as well as the total internal assets of each bank, which by [34] are assumed to be equal to the total internal liabilities of each bank. In <ref>, we have perturbed the total internal assets by adding some noise, and rebalancing them so that their total sum is invariant. The assets and liabilities can be approximated by exponential distributions. We report the distributions and the corresponding fits in  <ref>. The external assets and liabilities are linearly correlated with $R = 0.997$.

Venmo Dataset. The venmo dataset consists of $n = 7,178,381$ nodes and $m = 7,024,837$ edges (transactions) with an average (total) degree per user of 1.9572. The dataset consists of directed transactions from a sender node to a receiver node. There are no payment amounts data (for privacy reasons).  <ref> displays the degree distributions of the dataset and the corresponding power-law fits. The indegree distribution law is $y \propto x^{-2.3}$ and the outdegree distribution law is $y \propto x^{-3.7}$ with corresponding Pearson coefficients $-0.86$ and $-0.89$ respectively. We have computed all the weakly-connected components of the Venmo transaction network and kept 19 weakly-connected components which contained more than $100$ nodes. We generated fake transactions using Pareto laws.

[Total Internal Assets distribution with Exponential fit]
[External assets and external liabilities regression]
[External assets and liabilities distributions with Exponential fit]
[Equity distribution and Exponential fit]
EBA dataset statistics

[Indegree distribution]
[Outdegree distribution]

Venmo dataset degree distributions with power-law fits.

SafeGraph Data. We give the detailed steps we used and assumptions we relied on to build the SafeGraph data. We first describe the two different kinds of nodes, that form a bipartite graph where nodes of each kind lie in their corresponding set of the bipartition:

* POI nodes. We construct the POI nodes by setting a location on Earth and looking at the $k_{\mathrm{kNN}}$-nearest CBGs that contain POIs according to the Haversine distance. We choose the location  and use $k_{\mathrm{kNN}} = 3$.

* CBG nodes. We use the Monthly Patterns data for the period between April and May 2020 to determine the CBGs that interact with the POIs in question. For each POI, we use the column  to list the CBGs and the total number of visitors from each CBG to each POI. These visitors correspond to the number of unique devices that interact with the POI. Moreover, we assume that each distinct device represents a different person. For each CBG node we use data from the US Census we also log demographic characteristics such as: (i) race, (ii) average income, (iii) size of households, (iv) unemployment rates. Using these data for each CBG we calculate: (i) the probability of belonging to a racial minority group (i.e. non-White), (ii) the average income, (iii) the average size of a household on this CBG, (iv) the probability of someone in the CBG being unemployed.

The above network consists of 152 nodes (for our choice of parameters), but can be different if e.g. the starting coordinates or $k_{\mathrm{kNN}}$ are changed. We create the edges and the internal liabilities $p_{ij}$ as follows:

* We use the bucketed dwell times to determine the percentage of people that are working in a business. More specifically, we consider a worker a device that has spent more than 240 min in the POI and we classify the person as a non-worker otherwise. Workers come from CBGs and are paid wages whereas non-workers come from CBGs and have expenses on the corresponding POI. For every CBG $j$ let $p^{\mathrm{worker}}_j$ be this percentage.

* For every CBG $i$ that interacts with a POI $j$ with $n_{ij}$ people we add an edge $(j, i)$ referring to $n_{ij}^{\mathrm{workers}} = \lfloor n_{ij} \times p^{\mathrm{worker}}_j \rfloor$ workers, and an edge $(i, j)$ referring to $n_{ij}^{\mathrm{non-workers}} = n_{ij} - \lfloor n_{ij} \times p^{\mathrm{workers}}_j \rfloor $ non-worker nodes. We determine the weights of these edges (per unit) as follows: for the $(j, i)$ edge we take wage data by the US Economic Census and NAICS and create a liability by the business equal to the monthly wage of an employee which we multiply by $n_{ij}^{\mathrm{workers}}$. For the $(i, j)$ edge we use data from the Consumer Expenditure Survey conducted by the US Economic Census and add a liability regarding $n_{ij}^{\mathrm{non-workers}}$ equal to the average monthly expenditures due to the specific NAICS code of the business.

The number of workers for each POI $j$ is $n_j^{\mathrm{workers}} = \sum_{i \text{ is CBG}} n_{ij}^{\mathrm{workers}}$ (and the number of non-workers is defined respectively). Similarly we count the number of workers and number of non workers for each CBG $i$. We estimate the total number of households in the CBG that are related to interaction with the corresponding POIs[Of course the total number of people and households for each CBG is reported by the US Census the dataset would not be calibrated if we did not limit the transactions of each CBG with the corresponding POIs.]. We estimate the number of households as follows

\begin{equation*}
n_i^{\mathrm{households}} = \left \lceil \frac {\frac {n_i^{\mathrm{workers}}}  {p_i^{\mathrm{employed}}} + n_i^{\mathrm{non-workers}}} {\text{average size of household at CBG $i$}} \right \rceil .
\end{equation*}

The bailouts are determined as follows

* For each CBG $i$ we use the average income of the CBG and the average size of the household to calculate the value of a bailout as it would be imposed by the CARES act (see [1]). The bailout is multiplied by the estimated number of households calculated previously.

* For each POI $j$ we have data from loans that were given on multiple businesses as parts of the COVID-19 relief plan in the US and are available via SafeGraph. The loan value is normalized to span a month and is also normalized by the true number of jobs reported in each business and then is multiplied by the number of workers at POI $j$ in the bipartite network. The loan data also report the race of its owner (owners can also choose not to answer) so we can determine if the business is a minority or a non-minority-owned business. To fill the missing data we use the percentage of minority businesses as a real-valued score for the corresponding business.

Finally, we determine the external assets and liabilities

* For each POI $j$ we use the total assets earned annually from all establishments, normalized them by month,divide by the total number of workers for the specific NAICS code that the POI belongs, and multiply by the number of workers at the POI. Finally, from this amount we subtract the revenue due to nodes within the network (i.e. inbound edges) and take the positive part in case the result is negative. The external liabilities of the POI are determined in a similar way with the only change that the total weight of the outbound edges is subtracted in the end. To ensure that assumption:beta_max_lt_one holds throughout the experiments (and, thus, we have a unique equilibrium) we assert a minimum of $100 for the external liabilities. Such a number is smaller than the order of most quantities and thus does not significantly affect the results.

* For each CBG $i$ the process is similar where for external assets we use data regarding the average income as reported by the US Census, following a similar procedure as in the POI case, but now normalized with respect to the estimated number of households and the inbound connections from the POIs. Regarding the average expenses of a household, we the national US average of $\sim \! \$ 63,000$.

This completes the construction of the SafeGraph semi-artificial data experiment. It is important to note that all values are “normalized” with respect to entities participating in the network. This normalization process avoids scenaria like the following: there is a CBG where a small percentage of household heads are employed in the corresponding businesses so the weights of the network (internal liabilities) would have different scale than the external assets and liabilities resulting in diminishing network effects, namely a lot of liabilities are owned outside the network and hence the values of $\beta_i$ are very small and thus the effect of the relative internal liability matrix $A$ vanishes (i.e. $\beta_{\max} = \| A^T \|_1 \approx 0$) and thus the clearing payments become approximately $\bar p \approx b \wedge (c - x + L \odot\bar z)$; essentially eliminating network effects. In fact, without network effects, the problem is solvable in $O(n \log n)$ time: we calculate $\bar p_j' = b_j \wedge (c_j -x_j + L_j)$ in $O(n)$ time, we sort the values $v_j p_j'$ in decreasing order in $O(n \log n)$ time and then pick the first entries such that the budget constraint is respected. The correctness of this argument relies on the use of an elementary algebraic argument.

§ DATA ETHICS STATEMENT

All data in this paper comes from publicly available datasets derived and used in prior research.  The financial networks from banks can be found in the public domain and their corresponding publications. The Venmo network data are publicly available on GitHub, and do not contain transaction amounts. The mobility data were obtained from a public SafeGraph dataset in which privacy techniques have been applied by the data provider, and, thus, no human subjects are identifiable. Because all data consists of public, pre-existing datasets without identifiable individuals, the current work is exempt from IRB review.


# Urban Region Embedding via Multi-View Contrastive Prediction

Zechen Li1, Weiming Huang2, Kai Zhao3, Min Yang1, Yongshun Gong1, Meng Chen1
Corresponding author.

###### Abstract

Recently, learning urban region representations utilizing multi-modal data
(information views) has become increasingly popular, for deep understanding of
the distributions of various socioeconomic features in cities. However,
previous methods usually blend multi-view information in a posteriors stage,
falling short in learning coherent and consistent representations across
different views. In this paper, we form a new pipeline to learn consistent
representations across varying views, and propose the multi-view Contrastive
Prediction model for urban Region embedding (ReCP), which leverages the
multiple information views from point-of-interest (POI) and human mobility
data. Specifically, ReCP comprises two major modules, namely an intra-view
learning module utilizing contrastive learning and feature reconstruction to
capture the unique information from each single view, and inter-view learning
module that perceives the consistency between the two views using a
contrastive prediction learning scheme. We conduct thorough experiments on two
downstream tasks to assess the proposed model, i.e., land use clustering and
region popularity prediction. The experimental results demonstrate that our
model outperforms state-of-the-art baseline methods significantly in urban
region representation learning.

## Introduction

A deep understanding of the spatial distribution of various socioeconomic
factors in cities such as land use or population distribution, is important
for urban planning and management. In recent years, an increasingly popular
trend in the community of urban computing has been to partition a city into
numerous regions and utilize various urban sensory data to learn the latent
representations of the regions, which can subsequently be used in varying
urban sensing tasks, e.g., land usage clustering. house price prediction, and
population density inference (Liu et al. 2021; Li et al. 2022; Liu et al.
2023; Huang et al. 2023; Xu et al. 2023b; Li et al. 2023). This trend can also
be attributed to the prosperity of mobile sensing technologies, which has led
to the rapid accumulation of urban sensing data, such as human trajectories or
points-of-interest (POIs) (Chen, Yu, and Liu 2018; Zhang, Zhao, and Chen 2022;
Xu et al. 2023a; Zhang et al. 2023). Such various urban data provide more
opportunities for tackling the problem of region representation learning.

(a)  Figure 1: Illustration of (a) multi-view fusion paradigm and our proposed
(b) consistency learning paradigm for region embedding. In the right figure,
the solid and dotted rectangles denote the region representations
$\mathbf{Z}^{a}$ and $\mathbf{Z}^{m}$ from the attribute and mobility views,
respectively. The mutual information $I(\mathbf{Z}^{a},\mathbf{Z}^{m})$
(chartreuse area) quantifies the amount of information shared by
$\mathbf{Z}^{a}$ and $\mathbf{Z}^{m}$; the conditional entropy
$H(\mathbf{Z}^{a}|\mathbf{Z}^{m})$ (grey area) quantifies the amount of
information of $\mathbf{Z}^{a}$ conditioned on $\mathbf{Z}^{m}$. To learn
consistent region representations across different views, it is encouraged to
maximize $I(\mathbf{Z}^{a},\mathbf{Z}^{m})$ and minimize
$H(\mathbf{Z}^{a}|\mathbf{Z}^{m})$ and $H(\mathbf{Z}^{m}|\mathbf{Z}^{a})$.

Many previous studies have attempted to learn region representations by
utilizing human mobility data. For instance, Wang et al. (Wang and Li 2017)
construct flow graphs and spatial graphs using taxi flow data and propose a
graph embedding method to learn region representations. Yao et al. (Yao et al.
2018) extract human mobility patterns from taxi trajectories, and model the
co-occurrence of origin-destination regions to learn region representations.
The above methods merely rely on single-view data, which offers a limited
perspective of regions and fails to provide a comprehensive representation.
Further, recent studies (Zhang et al. 2021; Luo, Chung, and Chen 2022; Zhou et
al. 2023) propose learning region representations through integrating data in
multiple modalities, thus forming multiple information views. In this context,
the technical focus of recent region embedding studies has shifted towards the
fusion between multiple information views, where they usually follow the same
pipeline: separate single-view representation followed by multiple-view
fusion. Such a pipeline is demonstrated in Figure 1(a), where, it (1)
separately models each information view (usually with a graph structure) and
learns multiple single-view representations for each region, and (2) leverages
certain fusion techniques (e.g., based on attention mechanisms) to blend
multiple representations and yield the final multi-view region representation.

The previous multi-view region embedding methods have been effective in
certain analyses, but they come with a notable shortcoming: neglecting the
information consistency across different views when generating the final
region representation. Intuitively, the information carried by multiple views
of a region is highly correlated, and thus their representations should be
consistent. For example, an entertainment region could contain multiple bars
and restaurants (region attribute view based on POIs), as well as a large
number of nighttime mobility flows (human mobility view). Both views can
reflect the intrinsic characteristics of this region (i.e., entertainment
function). If we manage to leverage such correlation, it could be served as
the constraint during the process of learning representations for each view,
and enable the knowledge of transferring from one view to the other.
Ultimately, the multi-view representations would become highly consistent and
naturally fused.

Following the ideas above, we present a new pipeline - consistency learning
paradigm - for multi-view region embedding from an information theory
perspective (Tsai et al. 2021; Lin et al. 2021), where the multi-view
representations are naturally fused through exchanging information between
views along with learning view-specific region representations, rather than
treating fusion as a posterior process. This new pipeline is shown in Figure
1(b). Given two view-specific region representations $\mathbf{Z}^{a}$ and
$\mathbf{Z}^{m}$ (where they are from the region attribute view and the human
mobility view, respectively), we maximize the mutual information
$I(\mathbf{Z}^{a},\mathbf{Z}^{m})$ to increase the amount of the shared
information (consistency) in the region representations of the two views. We
also minimize the conditional entropy $H(\mathbf{Z}^{a}|\mathbf{Z}^{m})$ and
$H(\mathbf{Z}^{m}|\mathbf{Z}^{a})$ to diminish the inconsistent information
across the two views and improve the consistency further.

Based on the consistency learning paradigm, we propose a multi-view
Contrastive Prediction model for urban Region embedding (ReCP), which can
effectively enhance the consistency of region representations across different
views. ReCP consists of two major components: intra-view learning and inter-
view learning. In the intra-view learning component, to learn view-specific
region representations, we compare each region with other dissimilar ones to
embed the region into a latent space via contrastive learning; in the
meantime, we also utilize autoencoders to capture view-specific region
features for different views, which helps avoid model falling into a trivial
solution. In the inter-view learning component, to learn the cross-view
consistency of region representations, we design inter-view contrastive
learning by maximizing $I(\mathbf{Z}^{a},\mathbf{Z}^{m})$ and dual prediction
between views by minimizing $H(\mathbf{Z}^{a}|\mathbf{Z}^{m})$ and
$H(\mathbf{Z}^{m}|\mathbf{Z}^{a})$.

To summarize, our contributions are as follows:

  * •

We form a new pipeline following a consistency learning paradigm, to study the
urban region embedding problem by exploring the consistency across different
views, using both human mobility and POI data. Different from existing multi-
view region embedding methods which adopt the attention mechanisms to fuse
representations of different views, we propose to learn consistent multi-view
representations of regions by increasing the amount of shared information
across multiple views from the information entropy perspective.

  * •

We design the inter-view contrastive learning and dual prediction processes to
diminish the inconsistent information across views and learn an informative
and consistent region representation between different views, achieved by
maximizing the mutual information among different views and minimizing the
conditional entropy among them.

  * •

We conduct extensive experiments to evaluate our model with real-world
datasets. The results demonstrate that the proposed ReCP outperforms existing
methods on two downstream tasks by a margin. Data and source code are
available at https://anonymous.4open.science/r/ReCP.

## Problem Formulation

We formulate the urban region representation learning problem with the
following definitions:

######  Definition 1 (Urban Region)

A city can be partitioned into $n$ disjoint urban regions, denoted as
$\mathcal{R}=\left\\{r_{1},r_{2},...,r_{n}\right\\}$.

######  Definition 2 (Region Attributes)

In this study, region attributes are defined as inherent geographic features
of regions. Specifically, we consider Point of Interest (POI) categories as
region attributes following (Zhang, Long, and Cong 2022; Fu et al. 2019).
These region attributes are represented as a set
$\mathcal{A}=\\{\mathbf{A}_{1},\mathbf{A}_{2},\cdots,\mathbf{A}_{n}\\}$, where
$\mathbf{A}_{i}\in\mathbb{R}^{F}$ and $F$ represents the total number of POI
categories. Each dimension in $\mathbf{A}_{i}$ corresponds to the number of
POIs with a specific category in the region $r_{i}$.

######  Definition 3 (Human Mobility)

For a region $r_{i}$, we define its outflow feature $\mathbf{S}_{i}^{j,t}$ as
the number of trips made by all individuals originating from region $r_{i}$
and destined for region $r_{j}$ during a specific time interval $t$.
Consequently, we generate a collection of outflow features based on the
mobility data encompassing all regions within the set $\mathcal{R}$. This
collection is represented as
$\mathcal{S}=\\{\mathbf{S}_{1},\mathbf{S}_{2},\cdots,\mathbf{S}_{n}\\}$, where
$\mathbf{S}_{i}\in\mathbb{R}^{M}$. Here, $M$ is calculated as the product of
the number of regions, $n$, and the number of time intervals, $N_{t}$, within
a day, for instance, 24. Similarly, by considering $r_{i}$ as the destination
region and the other regions $r_{j}$ as the source regions, we can obtain an
inflow feature vector, denoted as $\mathbf{D}_{i}$, and finally obtain a
collection
$\mathcal{D}=\\{\mathbf{D}_{1},\mathbf{D}_{2},\cdots,\mathbf{D}_{n}\\}$ of
inflow features for all regions.

######  Problem 1 (Region Representation Learning)

Given the attribute features $\mathcal{A}$, outflow features $\mathcal{S}$,
and inflow features $\mathcal{D}$ of $n$ regions, our objective is to acquire
a collection of low-dimensional embeddings
$\mathcal{E}=\\{\mathbf{E}_{1},\mathbf{E}_{2},\cdots,\mathbf{E}_{n}\\}$, to
serve as the latent representation for each region.

Figure 2: The framework of ReCP.

## Methodology

The framework of ReCP is illustrated in Figure 2, which includes two major
components: 1) intra-view learning: for both region attribute and human
mobility view, it captures the representative features of each region by
intra-view contrastive learning to learn view-specific representations.
Additionally, feature reconstruction is designed within each view to recover
the original feature of the region, which helps avoid a trivial solution; 2)
inter-view learning: within the same region, it integrates representations
from different views through two learning objectives: inter-view contrastive
learning is used to enhance the consistency across different views, and dual
prediction is introduced to further diminish the inconsistent information
between views.

### Intra-view Learning

Initially, we learn view-specific region representations based on the region
attribute features $\mathcal{A}$ and the mobility features $\mathcal{S}$ and
$\mathcal{D}$, respectively. Within each view, we learn the latent
representation for each region by employing intra-view contrastive learning,
i.e., we compare each region with others to highlight distinctive features
within each region. Additionally, we design a within-view reconstruction loss
to avoid the trivial solution.

#### Intra-view Contrastive Learning

To learn region representations within each view, we design an intra-view
contrastive learning module, which compares each region with others. For a
given region $r_{i}$, we have three types of region features, including the
attribute feature $\mathbf{A}_{i}$, outflow feature $\mathbf{S}_{i}$, and
inflow feature $\mathbf{D}_{i}$. For simplicity, let $\mathbf{X}_{i}^{v}$
denote the raw feature for the $v$-th view. For a target region $r_{i}$, we
define its positive set as
$\mathcal{P}^{v}_{i}=\left\\{\mathbf{X}^{v}_{1},\mathbf{X}^{v}_{2},\cdots,\mathbf{X}^{v}_{K}\right\\}$,
where $\mathbf{X}^{v}_{1},\mathbf{X}^{v}_{2},\cdots,\mathbf{X}^{v}_{K}$ are
positive samples obtained through the data augmentation function following
(Zhang, Long, and Cong 2022), and $K$ is the number of positive samples. The
negative set $\mathcal{N}^{v}_{i}$ is defined as
$\mathcal{N}^{v}_{i}=\\{\mathbf{X}^{v}_{t}|t\neq i\\}$, which contains
features of regions except $r_{i}$.

We then map the raw features of regions into a latent representation,

$\mathbf{Z}_{i}^{v}=E^{\left(v\right)}(\mathbf{X}^{v}_{i}),$ (1)

where $E^{(v)}$ denotes the encoder for the $v$-th view. In practice, we
simply implement it as a fully connected neural network. As a result, we
obtain three types of region representations, $\mathbf{Z}^{a}_{i}$,
$\mathbf{Z}^{s}_{i}$ and $\mathbf{Z}^{d}_{i}$. Further, we compute the region
representation $\mathbf{Z}^{m}_{i}$ of the human mobility view as the average
of $\mathbf{Z}^{s}_{i}$ and $\mathbf{Z}^{d}_{i}$, i.e.,
$\mathbf{Z}^{m}_{i}=\left(\mathbf{Z}^{s}_{i}+\mathbf{Z}^{d}_{i}\right)/2$. To
maximize the similarity of positive pairs while minimizing the similarity of
negative pairs, the contrastive learning loss for the $v$-th view is defined
as follows,

$\displaystyle\mathcal{L}_{cl}^{v}=$
$\displaystyle\sum_{r_{i}\in\mathcal{R}}[-\log\sum_{k=1}^{K}{\exp(\frac{\mathbf{Z}_{i}^{v}\cdot\mathbf{Z}_{k}^{v}}{\tau})}+$
(2)
$\displaystyle\log(\sum_{k=1}^{K}{\exp(\frac{\mathbf{Z}_{i}^{v}\cdot\mathbf{Z}_{k}^{v}}{\tau})}+\sum_{t=1}^{|\mathcal{N}^{v}_{i}|}{\exp(\frac{\mathbf{Z}_{i}^{v}\cdot\mathbf{Z}_{t}^{v}}{\tau})})],$

where $\tau$ is the temperature parameter and $\mathcal{R}$ is the set of
regions. Further, the intra-view contrastive learning loss across all views is
formulated as

$\mathcal{L}_{cl}^{intra}=\mu\mathcal{L}_{cl}^{a}+\mathcal{L}_{cl}^{m}.$ (3)

where $\mu$ is the parameter controlling the balance between the attribute
view and the mobility view.

#### Intra-view Reconstruction

Given the feature $\mathbf{X}_{i}^{v}$ for the $v$-th view of the region
$r_{i}$, we further optimize the latent region representations via an
autoencoder and define the reconstruction loss $\mathcal{L}_{rec}^{v}$ as

$\mathcal{L}_{rec}^{v}=\sum_{r_{i}\in\mathcal{R}}{\left\|\mathbf{X}^{v}_{i}-D^{\left(v\right)}(E^{\left(v\right)}(\mathbf{X}^{v}_{i}))\right\|_{2}^{2}},$
(4)

where $E^{(v)}$ is the same as that in Equation (1) and $D^{(v)}$ is the
decoder for the $v$-th view to reconstruct the region features. Specifically,
we employ a fully connected network to implement $D^{(v)}$, which shares the
same number of layers and hidden sizes as $E^{(v)}$. Note that the autoencoder
structure is helpful to avoid the trivial solution.

The total reconstruction loss across all views is

$\mathcal{L}^{intra}_{rec}=\mu\mathcal{L}_{rec}^{a}+\mathcal{L}_{rec}^{m},$
(5)

where $\mu$ is the same weight parameter as that in Equation (3). So far, we
obtain two types of view-specific region representations $\mathbf{Z}_{i}^{a}$
and $\mathbf{Z}_{i}^{m}$ from the region attribute and human mobility views.

### Inter-view Learning

Different views of a region provide valuable information for describing the
region, often offering complementary insights. To learn consistent and
informative representations across different views, we employ inter-view
contrastive learning to improve collaboration and information exchange between
the views, achieved by maximizing the mutual information among different
views. Additionally, dual prediction between two views is leveraged to reduce
the impact of inconsistent information between the views by minimizing the
conditional entropy across them.

#### Inter-view Contrastive Learning

In the latent embedding space, we conduct contrastive learning to learn
consistent representations shared across different views, as recent
contrastive learning studies (He et al. 2020; Lin et al. 2021) have shown that
consistency could be learned by maximizing the mutual information of different
views. Formally, given the two representations $\mathbf{Z}_{i}^{a}$ and
$\mathbf{Z}_{i}^{m}$ of region $r_{i}$, we maximize the mutual information
between $\mathbf{Z}_{i}^{a}$ and $\mathbf{Z}_{i}^{m}$ from different views:

$\mathcal{L}^{inter}_{cl}=-\sum_{r_{i}\in\mathcal{R}}[I(\mathbf{Z}_{i}^{a},\mathbf{Z}_{i}^{m})+\alpha(H(\mathbf{Z}_{i}^{a})+H(\mathbf{Z}_{i}^{m}))],$
(6)

where $I(\cdot)$ represents mutual information, $H(\cdot)$ denotes information
entropy, and the parameter $\alpha$ controls the balance between mutual
information and information entropy. Note that the maximization of
$H\left(\mathbf{Z}_{i}^{a}\right)$ and $H\left(\mathbf{Z}_{i}^{m}\right)$ also
helps prevent trivial solutions where all regions are represented by the same
representation. Based on the definition of mutual information, $I(\cdot)$ is
defined as

$I\left(\mathbf{Z}^{a}_{i},\mathbf{Z}^{m}_{i}\right)=P\left(\mathbf{Z}^{a}_{i},\mathbf{Z}^{m}_{i}\right)log\frac{P\left(\mathbf{Z}^{a}_{i},\mathbf{Z}^{m}_{i}\right)}{P\left(\mathbf{Z}^{a}_{i}\right)P\left(\mathbf{Z}^{m}_{i}\right)},$
(7)

where $P\left(\mathbf{Z}^{a}_{i},\mathbf{Z}^{m}_{i}\right)$ represents the
joint probability distribution of $\mathbf{Z}^{a}_{i}$ and
$\mathbf{Z}^{m}_{i}$. To represent the joint probability distribution, we
employ a softmax function to transform the region representations
$\mathbf{Z}_{i}^{a}\in\mathbb{R}^{d}$ and
$\mathbf{Z}_{i}^{m}\in\mathbb{R}^{d}$ (where $d$ is the dimension of region
representations) with

$\mathbf{B}_{i}^{a}=\mathrm{softmax}\left(\mathbf{Z}_{i}^{a}\right),\mathbf{B}_{i}^{m}=\mathrm{softmax}\left(\mathbf{Z}_{i}^{m}\right),$
(8)

where $\mathbf{B}_{i}^{a}\in\mathbb{R}^{d}$ and
$\mathbf{B}_{i}^{m}\in\mathbb{R}^{d}$ can be interpreted as the probability
distributions. Considering the entire set $\mathcal{R}$ containing $n$
regions, we define the matrix $\mathbf{M}\in\mathbb{R}^{d\times d}$ as the
joint probability distribution of $\mathbf{Z}^{a}$ and $\mathbf{Z}^{m}$,

$\mathbf{M}=\frac{1}{n}\sum^{n}_{i=1}{\mathbf{B}_{i}^{a}\left(\mathbf{B}_{i}^{m}\right)}^{\mathrm{T}}.$
(9)

We denote the element located at the $r$-th row and the $r\prime$-th column of
the matrix as $\mathbf{M}_{rr\prime}$, and the sum of the elements in matrix
$\mathbf{M}$ along the $r$-th row (the $r\prime$-th column) as
$\mathbf{M}_{r}$ ($\mathbf{M}_{r\prime}$). $\mathbf{M}_{rr\prime}$ represents
the joint probability, while $\mathbf{M}_{r}$ and $\mathbf{M}_{r\prime}$
represent the marginal probability, respectively. Then we could compute the
mutual information $I\left(\mathbf{Z}^{a},\mathbf{Z}^{m}\right)$ as follows,

$I\left(\mathbf{Z}^{a},\mathbf{Z}^{m}\right)=\sum^{d}_{r=1}{\sum^{d}_{r\prime=1}{\mathbf{M}_{rr\prime}\log\frac{\mathbf{M}_{rr\prime}}{\mathbf{M}_{r}\cdot\mathbf{M}_{r\prime}}}}.$
(10)

Information entropy $H(\mathbf{Z}^{v}_{i})$ is defined as follows,

$H(\mathbf{Z}^{v}_{i})=-P(\mathbf{Z}^{v}_{i})log{P(\mathbf{Z}^{v}_{i})},$ (11)

where $v\in\\{a,m\\}$. Following the above definition of $\mathbf{M}$,
$H(\mathbf{Z}^{v}_{i})$ could be computed as

$\displaystyle H(\mathbf{Z}^{a})$
$\displaystyle=-\sum^{d}_{r=1}{\mathbf{M}_{r}\log{\mathbf{M}_{r}}},$ (12)
$\displaystyle H(\mathbf{Z}^{m})$
$\displaystyle=-\sum^{d}_{r\prime=1}{\mathbf{M}_{r\prime}\log{\mathbf{M}_{r\prime}}}.$

Combining Equations (6), (10), and (12), the inter-view contrastive learning
loss is formulated as

$\mathcal{L}^{inter}_{cl}=-\sum^{d}_{r=1}{\sum^{d}_{r\prime=1}{\mathbf{M}_{rr\prime}\ln\frac{\mathbf{M}_{rr\prime}}{\mathbf{M}_{r}^{\alpha+1}\cdot\mathbf{M}_{r\prime}^{\alpha+1}}}}.$
(13)

where $\alpha$ is the weight parameter defined in the Equation (6).

#### Inter-view Dual Prediction

To further diminish the inconsistency across different views, we predict the
view-specific region representation by minimizing the conditioned entropy.
Formally, given the region representations $\mathbf{Z}^{a}$ and
$\mathbf{Z}^{m}$, we minimize the conditional entropy
$H(\mathbf{Z}^{p}|\mathbf{Z}^{q})$, where $p=a,q=m$ or $p=m,q=a$. On one hand,
$\mathbf{Z}^{q}$ contains nearly all the information required to represent the
$p$-th view if $\mathbf{Z}^{q}$ can perfectly predict $\mathbf{Z}^{p}$ for any
$(\mathbf{Z}^{p},\mathbf{Z}^{q})\sim{P}_{\mathbf{Z}^{p},\mathbf{Z}^{q}}$. On
the other hand, $\mathbf{Z}^{q}$ diminishes the inconsistent information
within the $q$-th view if $\mathbf{Z}^{p}$ can perfectly predict
$\mathbf{Z}^{q}$ under the constraint where $I(\mathbf{Z}^{p},\mathbf{Z}^{q})$
is maximized. Mathematically, $H(\mathbf{Z}^{p}|\mathbf{Z}^{q})$ is defined as

$H\left(\mathbf{Z}^{p}\left|\mathbf{Z}^{q}\right.\right)=-{\mathbb{E}_{{P}}}_{{}_{\mathbf{Z}^{p},\mathbf{Z}^{q}}}\left[\log{P}\left(\mathbf{Z}^{p}|\mathbf{Z}^{q}\right)\right].$
(14)

To minimize $H\left(\mathbf{Z}^{p}\left|\mathbf{Z}^{q}\right.\right)$, a
common approach is to assume a variational distribution
${Q}\left(\mathbf{Z}^{p}|\mathbf{Z}^{q}\right)$ for $\mathbf{Z}^{p}$ and
$\mathbf{Z}^{q}$. Specially, we present to maximize
${\mathbb{E}_{{P}}}_{{}_{\mathbf{Z}^{p},\mathbf{Z}^{q}}}\left[\log{Q}\left(\mathbf{Z}^{p}|\mathbf{Z}^{q}\right)\right]$,
which serves as a lower bound of
${\mathbb{E}_{{P}}}_{{}_{\mathbf{Z}^{p},\mathbf{Z}^{q}}}\left[\log{P}\left(\mathbf{Z}^{p}|\mathbf{Z}^{q}\right)\right]$.
${Q}\left(\cdot|\cdot\right)$ can be any distribution such as Gaussian or
Laplacian. In this work, we simply adopt the Gaussian distribution
${N}\left.(\mathbf{Z}^{p}|F^{\left(q\right)}\left(\mathbf{Z}^{q}\right),\sigma\mathbf{I}\right.)$,
where $F^{\left(q\right)}\left(\cdot\right)$ represents a parameterized
function mapping $\mathbf{Z}^{q}$ to $\mathbf{Z}^{p}$, and $\sigma\mathbf{I}$
denotes the variance matrix. By ignoring the constants derived from the
Gaussian distribution, maximizing
${\mathbb{E}_{{P}}}_{{}_{\mathbf{Z}^{p},\mathbf{Z}^{q}}}\left[\log{Q}\left(\mathbf{Z}^{p}|\mathbf{Z}^{q}\right)\right]$
is equivalent to minimizing

${\mathbb{E}_{{P}}}_{{}_{\mathbf{Z}^{p},\mathbf{Z}^{q}}}\left\|\mathbf{Z}^{p}-F^{(q)}\left(\mathbf{Z}^{q}\right)\right\|^{2}_{2}.$
(15)

Then the dual prediction loss can be formulated as

$\mathcal{L}_{dp}^{inter}=\sum_{r_{i}\in\mathcal{R}}{\left\|\mathbf{Z}_{i}^{m}-F^{\left(a\right)}\left(\mathbf{Z}_{i}^{a}\right)\right\|^{2}_{2}+\left\|\mathbf{Z}_{i}^{a}-F^{\left(m\right)}\left(\mathbf{Z}_{i}^{m}\right)\right\|^{2}_{2}}.$

Here, $F^{\left(a\right)}$ and $F^{\left(m\right)}$ are respectively
implemented as fully-connected networks, with each layer followed by a batch
normalization layer and a ReLU layer. Note that the above loss may lead to
model collapse without the intra-view reconstruction loss (Equation (4)),
i.e., $\mathbf{Z}_{i}^{a}$ and $\mathbf{Z}_{i}^{m}$ from different views
become equivalent to the same constant.

Finally, the inter-view learning loss is defined as

$\mathcal{L}_{inter}=\mathcal{L}_{dp}^{inter}+\mathcal{L}_{cl}^{inter}.$ (16)

### Model Training

The final objective function is defined as

$\mathcal{L}=\mathcal{L}_{inter}+\lambda_{1}\mathcal{L}^{intra}_{cl}+\lambda_{2}\mathcal{L}^{intra}_{rec},$
(17)

where $\lambda_{1}$ and $\lambda_{2}$ are parameters controlling the weights
of different losses. After learning the latent representations
$\mathbf{Z}^{a}$ and $\mathbf{Z}^{m}$, we simply concatenate them as the final
multi-view region representation, i.e.,
$\mathbf{E}_{i}=\mathbf{Z}^{a}_{i}\left|\right|\mathbf{Z}^{m}_{i}$.

## Experiments

We start by presenting the experimental settings, followed by the evaluation
of the learned region representations on two popular downstream tasks: land
use clustering and region popularity prediction.

### Experimental Settings

Datasets. We collect a diverse set of real-world data from NYC Open
Data111https://opendata.cityofnewyork.us and use the Manhattan borough as the
study area. We partition Manhattan into 270 regions based on the city
boundaries designed by the US Census
Bureau222https://www.census.gov/data.html. As for the human mobility data, we
employ complete taxi trip records from February 2014 as our training data. We
utilize the NYC check-in and POI data provided by (Yang et al. 2014) for our
model training and the popularity prediction task. The detailed description of
datasets is shown in Table 1. Based on these data, we construct the region
features including $\mathcal{A}$, $\mathcal{S}$, and $\mathcal{D}$ for model
training.

Table 1: Data description (K=$10^{3}$, M=$10^{6}$). Dataset | Description
---|---
Regions |  270 regions divided by streets in Manhattan
Taxi trips |  10M taxi trips during February, 2014
POI data |  10K POIs with 244 categories
Check-in data |  100K check-in records
Table 2: Performance comparison on two downstream tasks, where the performance
improvements of ReCP are compared with the best of these baseline methods,
marked by the asterisk.

Method | Land Usage Clustering | Region Popularity Prediction
---|---|---
NMI | ARI | F-measure | MAE | RMSE | R2
HDGE | 0.469 ± 0.01 | 0.095 ± 0.01 | 0.117 ± 0.01 | 334.43 ± 10.17 | 474.94 ± 9.49 | 0.079 ± 0.04
ZE-Mob | 0.437 ± 0.02 | 0.071 ± 0.01 | 0.097 ± 0.01 | 282.42 ± 13.71 | 418.02 ± 12.69 | 0.286 ± 0.04
MV-PN | 0.407 ± 0.01 | 0.036 ± 0.01 | 0.070 ± 0.01 | 291.17 ± 16.54 | 435.23 ± 16.52 | 0.226 ± 0.06
CGAL | 0.414 ± 0.08 | 0.059 ± 0.06 | 0.091 ± 0.06 | 351.10 ± 51.20 | 486.96 ± 52.58 | 0.021 ± 0.20
MVURE | 0.735 ± 0.01 | 0.400 ± 0.02 | 0.415 ± 0.02 | 236.25 ± 7.86 | 347.01 ± 11.70 | 0.508 ± 0.03
MGFN | 0.748 ± 0.01 | 0.424 ± 0.03 | 0.437 ± 0.03 | 240.37 ± 11.99 | 354.24 ± 17.14 | 0.487 ± 0.05
ReMVC | 0.761* ± 0.02 | 0.455* ± 0.04 | 0.462* ± 0.04 | 283.02 ± 18.03 | 406.25 ± 18.00 | 0.325 ± 0.06
HREP | 0.757 ± 0.01 | 0.448 ± 0.03 | 0.457 ± 0.03 | 217.52* ± 10.98 | 318.41* ± 14.54 | 0.585* ± 0.04
ReCP | 0.780 ± 0.01 | 0.483 ± 0.01 | 0.499 ± 0.02 | 195.16 ± 18.70 | 291.19 ± 20.04 | 0.652 ± 0.05
Improvements | 2.50% | 6.15% | 8.01% | 10.28% | 8.55% | 11.45%

Model Parameters. In our experiments, the dimension of region representations
is set to 96. In the intra-view reconstruction module, we set the number of
layers at 3 and the hidden size at 128 for the encoder $E^{(v)}$ and decoder
$D^{(v)}$; in the intra-view contrastive learning module, following the
settings in (Zhang, Long, and Cong 2022), we set the number of positive
samples for region attribute and human mobility data at 3 and 4, and the
parameter $\mu$ controlling the balance between different views at 0.0001. In
the inter-view dual prediction module, we set the number of layers at 3 and
the hidden size at 96 for $F^{(a)}$ and $F^{(m)}$; in the inter-view
contrastive learning module, we set the parameter $\alpha$ at 9. We set the
hyper-parameters $\lambda_{1}$ and $\lambda_{2}$ in the final objective loss
at 1. Note that the optimal model parameters are selected using grid search
with a small but adaptive step size. To optimize our model, we adopt Adam and
initialize the learning rate at 0.01 with a linear decay.

Baselines. We compare the performance of ReCP with several state-of-the-art
region embedding methods.

  * •

HDGE. (Wang and Li 2017) constructs flow graphs and spatial graphs using taxi
data and learns region representations with graph embedding methods.

  * •

ZE-Mob. (Yao et al. 2018) models co-occurrence patterns between regions from
mobility data to learn region representations.

  * •

MV-PN. (Fu et al. 2019) models both inter-region and intra-region information
to construct multi-view POI-POI networks within each region.

  * •

CGAL. (Zhang et al. 2019) extends MV-PN and incorporates the spatial structure
and spatial autocorrelation among regions to learn region representations.

  * •

MVURE. (Zhang et al. 2021) learns region representations by cross-view
information sharing and multi-view fusion with human mobility and region
attributes.

  * •

MGFN. (Wu et al. 2022) designs multi-level cross-attention mechanisms to
extract region representations from multiple mobility patterns.

  * •

ReMVC. (Zhang, Long, and Cong 2022) learns region representations through both
intra-view and inter-view contrastive learning modules.

  * •

HREP. (Zhou et al. 2023) constructs heterogeneous graphs and uses relation-
aware graph embedding to learn region representations.

### Land Usage Clustering

We use the district division by the community boards (Berg 2007) as ground
truth and divide the Manhattan borough into 29 districts, following the
settings in (Zhang, Long, and Cong 2022). We cluster regions into groups by
$k$-means clustering ($k=29$), using region representations as inputs. The
regions with the same land usage type are expected to be assigned to the same
cluster. The experimental results are evaluated using three metrics:
Normalized Mutual Information (NMI), Adjusted Rand Index (ARI), and F-measure
following (Yao et al. 2018; Zhang et al. 2021). We assess all the methods
using the same dataset and conduct 10 runs to report the mean value with the
standard deviation in Table 2. From the results, we observe that:

  * •

HDGE and ZE-Mob exhibit relatively inferior performance as they merely model
co-occurrence patterns using human mobility data. MGFN demonstrates better
performance than HDGE and ZE-Mob, as it designs a deep model based on cross-
attention mechanisms to capture complex mobility patterns from spatial-
temporal human mobility data.

  * •

The methods that model multi-view information generally achieve satisfactory
results, validating the importance of effectively integrating multi-view
information for region embedding. Specifically, MV-PN and CGAL exhibit poor
performance as they simply combine region representations from two views,
lacking the deep interaction between views; MVURE and HREP design attention-
based mechanisms to fuse the multi-view information, consequently yielding
superior performance; ReMVC adopts contrastive learning to model intra-view
and inter-view information and also obtains good results.

  * •

The proposed ReCP outperforms all these baselines, as it explores the
consistency across different views in region embedding. Compared with ReMVC,
ReCP achieves average improvements of 2.50%, 6.15%, and 8.01% in terms of NMI,
ARI, and F-measure, respectively. Moreover, the results of the superiority
paired t-test indicate that the improvement of ReCP over the baselines is
statistically significant, with a $p$-value less than 0.01.

### Region Popularity Prediction

Another commonly-compared downstream task to evaluate the region
representations is popularity prediction, where we aggregate the check-in
counts within each region as the ground truth of popularity following (Yang et
al. 2014; Zhang, Long, and Cong 2022). We take region representations as input
and train the Lasso regression model. The evaluation results including Mean
Absolute Error (MAE), Root Mean Square Error (RMSE), and Coefficient of
Determination (R2) are obtained by 5-fold cross-validation, as reported in
Table 2. From the results, we observe that the multi-view fusion methods
including MVURE and HREP achieve decent performance, which further validates
the necessity of integrating multi-view information in region embedding. ReCP
performs the best among all methods, e.g., compared to HREP, ReCP achieves
average improvements of 10.28%, 8.55%, and 11.45% in terms of MAE, RMSE, and
R2. These results indicate that it is an effective way to learn better region
representations by utilizing the new pipeline following the consistency
learning paradigm.

### Ablation Study and Parameter Analysis

#### Ablation study

We design four variants to explore how each module of ReCP affects the
clustering and regression performance:

  * •

ReCP w/o CL: we remove the intra-view contrastive learning loss.

  * •

ReCP w/o Rec: we remove the intra-view reconstruction loss and only use the
encoder to extract features.

  * •

ReCP w/o IV: we remove the inter-view learning module and directly concatenate
region representations from the two views without the constraint of
consistency learning.

  * •

ReCP w/o DP: we remove the inter-view dual prediction loss.

From the results in Figure 3, we observe that:

1) ReCP w/o CL achieves the lowest performance in both tasks, indicating that
the intra-view contrastive learning loss is a crucial component in our model
for learning view-specific feature representations of regions.

2) ReCP w/o Rec achieves worse performance than ReCP, supporting the
aforementioned claim that the intra-view reconstruction loss could help
prevent the model from converging to a trivial solution.

3) ReCP demonstrates an improvement of 29.84% (in terms of ARI) and 4.00% (in
terms of R2) when compared to ReCP w/o IV. This finding suggests that the
proposed inter-view learning module effectively leverages the multi-view
information and highlights the importance of consistency learning across
different views.

4) ReCP w/o DP outperforms ReCP w/o IV but performs worse than ReCP,
indicating that both the inter-view contrastive learning loss (which maximizes
the mutual information between views) and the inter-view dual prediction loss
(which minimizes the conditional entropy across them) are important for
learning multi-view region representations.

(a) Land Usage Clustering

(b) Region Popularity Prediction

Figure 3: Performance comparison of different modules.

#### Parameter sensitivity

The parameters $\lambda_{1}$ and $\lambda_{2}$ govern the weighting of various
losses. We vary their values within the range of $\\{0.01,0.1,1,10,100\\}$ to
assess the impact of $\lambda_{1}$ and $\lambda_{2}$ on the model performance.
As depicted in Figure 4, ReCP achieves satisfactory performance when we set
both $\lambda_{1}$ and $\lambda_{2}$ at 1.

(a) Land Usage Clustering

(b) Region Popularity Prediction

Figure 4: Parameter analysis on both downstream tasks.

## Related Work

Traditional methods for region embedding typically utilize human mobility data
to analyze the transition patterns between urban regions. These methods are
often based on the word2vec framework and learn the latent representations of
regions (Wang and Li 2017; Yao et al. 2018). In a similar vein, Wu et al. (Wu
et al. 2022) incorporate mobility graphs with spatio-temporal similarity as
mobility patterns, and propose multi-level cross-attention mechanisms to
extract comprehensive region representations from these patterns.
Additionally, some studies focus on leveraging the inherent attributes of
regions to learn latent representations. For instance, Zhang et al. (Zhang et
al. 2019) construct multiple spatial graphs to represent the geographic
structure of regions. By transforming the region embedding problem into a
graph embedding problem, they primarily capture the spatial structure within
regions and the spatial autocorrelation between regions. Another approach,
proposed by Wang et al. (Wang, Li, and Rajagopal 2020), involves mining street
views and textual information of POIs within regions to learn representations.

Moreover, there have been studies that learn region representations by
incorporating both attribute features within regions and mobility patterns
between regions. For instance, Fu et al. (Fu et al. 2019) propose an
autoencoder framework that effectively captures inter-region correlations and
intra-region structural information during the process of region embedding.
Zhang et al. (Zhang et al. 2021) model multi-view region correlations by
leveraging human mobility data and inherent region attributes, and employ a
graph attention mechanism to acquire region representations from each view of
the established correlations. Zhou et al. (Zhou et al. 2023) learn relation-
specific region representations from various types of relations in a
heterogeneous graph constructed using human mobility, POI data, and geographic
neighbors of regions. They devise an attention-based fusion technique to
integrate shared information among different types of correlations.
Additionally, Zhang et al. (Zhang, Long, and Cong 2022) introduce a multi-view
region embedding model based on contrastive learning, which incorporates an
intra-view contrastive learning module to discern distinct representations and
an inter-view contrastive learning module to facilitate the transfer of
knowledge across multiple views.

## Conclusion

In this paper, we form a new pipeline based on the consistency learning
paradigm for multi-view region embedding. Under the hood, we propose a multi-
view Contrastive Prediction model for urban Region embedding (ReCP) by
exploring the consistency across two views, leveraging both POI and human
mobility data. The ReCP model consists of two modules: an intra-view learning
module that utilizes contrastive learning and feature reconstruction which
learn region representations specific to each view, and an inter-view learning
module utilizing a contrastive prediction learning scheme that enhances the
consistency between two views. To evaluate the effectiveness of our proposed
model, we conduct comprehensive experiments on two downstream tasks: land use
clustering and region popularity prediction. The experimental results
demonstrate that the proposed ReCP model outperforms state-of-the-art
embedding methods, proving that retaining consistency across views is pivotal
for effective region embedding.

## References

  * Berg (2007) Berg, B. F. 2007.  _New York City Politics: Governing Gotham_.  Rutgers University Press.
  * Chen, Yu, and Liu (2018) Chen, M.; Yu, X.; and Liu, Y. 2018.  PCNN: Deep convolutional networks for short-term traffic congestion prediction.  _IEEE Transactions on Intelligent Transportation Systems_ , 19(11): 3550–3559.
  * Fu et al. (2019) Fu, Y.; Wang, P.; Du, J.; Wu, L.; and Li, X. 2019.  Efficient region embedding with multi-view spatial networks: A perspective of locality-constrained spatial autocorrelations.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 33, 906–913.
  * He et al. (2020) He, K.; Fan, H.; Wu, Y.; Xie, S.; and Girshick, R. 2020.  Momentum contrast for unsupervised visual representation learning.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 9729–9738.
  * Huang et al. (2023) Huang, W.; Zhang, D.; Mai, G.; Guo, X.; and Cui, L. 2023.  Learning urban region representations with POIs and hierarchical graph infomax.  _ISPRS Journal of Photogrammetry and Remote Sensing_ , 196: 134–145.
  * Li et al. (2022) Li, T.; Xin, S.; Xi, Y.; Tarkoma, S.; Hui, P.; and Li, Y. 2022.  Predicting multi-level socioeconomic indicators from structural urban imagery.  In _Proceedings of the 31st ACM International Conference on Information & Knowledge Management_, 3282–3291.
  * Li et al. (2023) Li, Y.; Huang, W.; Cong, G.; Wang, H.; and Wang, Z. 2023.  Urban Region Representation Learning with OpenStreetMap Building Footprints.  In _Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining_ , 1363–1373.
  * Lin et al. (2021) Lin, Y.; Gou, Y.; Liu, Z.; Li, B.; Lv, J.; and Peng, X. 2021.  Completer: Incomplete multi-view clustering via contrastive prediction.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , 11174–11183.
  * Liu et al. (2021) Liu, C.; Yang, Y.; Yao, Z.; Xu, Y.; Chen, W.; Yue, L.; and Wu, H. 2021.  Discovering urban functions of high-definition zoning with continuous human traces.  In _Proceedings of the 30th ACM International Conference on Information & Knowledge Management_, 1048–1057.
  * Liu et al. (2023) Liu, Y.; Zhang, X.; Ding, J.; Xi, Y.; and Li, Y. 2023.  Knowledge-infused contrastive learning for urban imagery-based socioeconomic prediction.  In _Proceedings of the ACM Web Conference 2023_ , 4150–4160.
  * Luo, Chung, and Chen (2022) Luo, Y.; Chung, F.-l.; and Chen, K. 2022.  Urban region profiling via multi-graph representation learning.  In _Proceedings of the 31st ACM International Conference on Information & Knowledge Management_, 4294–4298.
  * Tsai et al. (2021) Tsai, Y.-H.; Wu, Y.; Salakhutdinov, R.; and Morency, L.-P. 2021.  Self-supervised learning from a multi-view perspective.  In _Proceedings of the International Conference on Learning Representations, 2021_.
  * Wang and Li (2017) Wang, H.; and Li, Z. 2017.  Region representation learning via mobility flow.  In _Proceedings of the 2017 ACM on Conference on Information and Knowledge Management_ , 237–246.
  * Wang, Li, and Rajagopal (2020) Wang, Z.; Li, H.; and Rajagopal, R. 2020.  Urban2vec: Incorporating street view imagery and pois for multi-modal urban neighborhood embedding.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 34, 1013–1020.
  * Wu et al. (2022) Wu, S.; Yan, X.; Fan, X.; Pan, S.; Zhu, S.; Zheng, C.; Cheng, M.; and Wang, C. 2022\.  Multi-graph fusion networks for urban region embedding.  In _Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence_ , 2312–2318.
  * Xu et al. (2023a) Xu, R.; Chen, M.; Gong, Y.; Liu, Y.; Yu, X.; and Nie, L. 2023a.  TME: Tree-guided multi-task embedding learning towards semantic venue annotation.  _ACM Transactions on Information Systems_ , 41(4).
  * Xu et al. (2023b) Xu, R.; Huang, W.; Zhao, J.; Chen, M.; and Nie, L. 2023b.  A spatial and adversarial representation learning approach for land use classification with POIs.  _ACM Transactions on Intelligent Systems and Technology_ , 14(6): 1–25.
  * Yang et al. (2014) Yang, D.; Zhang, D.; Zheng, V. W.; and Yu, Z. 2014.  Modeling user activity preference by leveraging user spatial temporal characteristics in LBSNs.  _IEEE Transactions on Systems, Man, and Cybernetics: Systems_ , 45(1): 129–142.
  * Yao et al. (2018) Yao, Z.; Fu, Y.; Liu, B.; Hu, W.; and Xiong, H. 2018.  Representing urban functions through zone embedding with human mobility patterns.  In _Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence_.
  * Zhang, Zhao, and Chen (2022) Zhang, C.; Zhao, K.; and Chen, M. 2022.  Beyond the limits of predictability in human mobility prediction: context-transition predictability.  _IEEE Transactions on Knowledge and Data Engineering_ , 35(5): 4514–4526.
  * Zhang et al. (2023) Zhang, D.; Xu, R.; Huang, W.; Zhao, K.; and Chen, M. 2023.  Towards an integrated view of semantic annotation for POIs with spatial and textual information.  In _Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence_ , 2441–2449.
  * Zhang, Long, and Cong (2022) Zhang, L.; Long, C.; and Cong, G. 2022.  Region embedding with intra and inter-view contrastive learning.  _IEEE Transactions on Knowledge and Data Engineering_.
  * Zhang et al. (2021) Zhang, M.; Li, T.; Li, Y.; and Hui, P. 2021.  Multi-view joint graph representation learning for urban region embedding.  In _Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence_ , 4431–4437.
  * Zhang et al. (2019) Zhang, Y.; Fu, Y.; Wang, P.; Li, X.; and Zheng, Y. 2019.  Unifying inter-region autocorrelation and intra-region structures for spatial embedding via collective adversarial learning.  In _Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, 1700–1708.
  * Zhou et al. (2023) Zhou, S.; He, D.; Chen, L.; Shang, S.; and Han, P. 2023.  Heterogeneous region embedding with prompt learning.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 37, 4981–4989.