A Consistent Diffusion-Based Algorithm for Semi-Supervised Graph Learning

Abstract

The task of semi-supervised classification aims at assigning labels to all nodes of a graph based on the labels known for a few nodes, called the seeds. One of the most popular algorithms relies on the principle of heat diffusion, where the labels of the seeds are spread by thermo-conductance and the temperature of each node at equilibrium is used as a score function for each label. In this paper, we prove that this algorithm is not consistent unless the temperatures of the nodes at equilibrium are centered before scoring. This crucial step does not only make the algorithm provably consistent on a block model but brings significant performance gains on real graphs.

Appendices

Appendix

A Proof of Lemma 1

Proof

In view of (2), we have:

$$\begin{aligned} (n_1(p-q) + nq) T_1 &= s_1 p + (n_1-s_1)pT_1 + \sum _{j\ne 1} (n_j - s_j) qT_j,\\ (n_k(p-q) + nq) T_k& = s_1 q + (n_k-s_k)pT_k + \sum _{j\ne k} (n_j - s_j) qT_j, \end{aligned}$$

for $k=2,\ldots ,K$. We deduce:

$$\begin{aligned} (s_1(p-q) + nq) T_1 &= s_1 p + Uq,\\ (s_k(p-q) + nq) T_k &= s_1 q + Uq\quad \quad \forall k=2,\ldots ,K, \end{aligned}$$

with

$$ U = \sum _{j=1}^K (n_j - s_j) T_j. $$

The proof then follows from the fact that

$$ n \bar{T} = s_1 + \sum _{j=1}^K (n_j - s_j) T_j = s_1 + U. $$

B Proof of Theorem 1

Proof

Let $\varDelta ^{(1)}_k = T_k - \bar{T}$ be the deviation of temperature of non-seed nodes of block k for the Dirichlet problem associated with label 1. In view of Lemma 1, we have:

$$\begin{aligned} (s_1(p-q) + nq) \varDelta ^{(1)}_1 &= s_1 (p-q) (1-\bar{T}),\\ (s_k(p-q) + nq)\varDelta ^{(1)}_k &= -s_k(p-q) \bar{T} \quad \quad k=2,\ldots ,K, \end{aligned}$$

For $p>q$, using the fact that $\bar{T} \in (0,1)$, we get $\varDelta ^{(1)}_1 > 0$ and $\varDelta ^{(1)}_k<0$ for all $k=2,\ldots ,K$. By symmetry, for each label $l = 1,\ldots ,K$, $\varDelta ^{(l)}_l > 0$ and $\varDelta ^{(l)}_k<0$ for all $k\ne l$. We deduce that for each block k, $\hat{y}_i=\arg \max _{l}\varDelta ^{(l)}_k = k$ for each free node i of block k.