Compared with the rich studies on the motor brain-computer interface (BCI), the recently emerging affective BCI presents distinct challenges since the brain functional connectivity networks involving emotion are not well investigated. Previous studies on emotion recognition based on electroencephalography (EEG) signals mainly rely on single-channel-based feature extraction methods. In this paper, we propose a novel emotion-relevant critical subnetwork selection algorithm and investigate three EEG functional connectivity network features: strength, clustering coefficient, and eigenvector centrality. The discrimination ability of the EEG connectivity features in emotion recognition is evaluated on three public emotion EEG datasets: SEED, SEED-V, and DEAP. The strength feature achieves the best classification performance and outperforms the state-of-the-art differential entropy feature based on single-channel analysis. The experimental results reveal that distinct functional connectivity patterns are exhibited for the five emotions of disgust, fear, sadness, happiness, and neutrality. Furthermore, we construct a multimodal emotion recognition model by combining the functional connectivity features from EEG and the features from eye movements or physiological signals using deep canonical correlation analysis. The classification accuracies of multimodal emotion recognition are 95.08/6.42% on the SEED dataset, 84.51/5.11% on the SEED-V dataset, and 85.34/2.90% and 86.61/3.76% for arousal and valence on the DEAP dataset, respectively. The results demonstrate the complementary representation properties of the EEG connectivity features with eye movement data. In addition, we find that the brain networks constructed with 18 channels achieve comparable performance with that of the 62-channel network in multimodal emotion recognition and enable easier setups for BCI systems in real scenarios.