Quasi-kernels in split graphs

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The ``small quasi-kernel conjecture,'' proposed by Erd\H{o}s and Sz\'ekely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a $1/2$ ratio, but even with larger ratio, this property is known to hold only for few classes of graphs. The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph $D$ has a quasi-kernel of size at most $\frac{2}{3}|V(D)|$, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard

On the Small Quasi-kernel conjecture

The Chv\'atal-Lov\'asz theorem from 1974 establishes in every finite digraph $G$ the existence of a quasi-kernel, i.e., an independent $2$-out-dominating vertex set. In the same spirit, the Small Quasi-kernel Conjecture, proposed by Erd\H{o}s and Sz\'ekely in 1976, asserts the existence of a quasi-kernel of order at most $|V(G)|/2$ if $G$ does not have sources. Despite repeated efforts, the conjecture remains wide open. This work contains a number of new results towards the conjecture. In our main contribution we resolve the conjecture for all directed graphs without sources containing a kernel in the second out-neighborhood of a quasi-kernel. Furthermore, we provide a novel strongly connected example demonstrating the asymptotic sharpness of the conjecture. Additionally, we resolve the conjecture in a strong form for all directed unicyclic graphs.Comment: 12 pages, 1 figur

A Multi-view Context-aware Approach to Android Malware Detection and Malicious Code Localization

Existing Android malware detection approaches use a variety of features such as security sensitive APIs, system calls, control-flow structures and information flows in conjunction with Machine Learning classifiers to achieve accurate detection. Each of these feature sets provides a unique semantic perspective (or view) of apps' behaviours with inherent strengths and limitations. Meaning, some views are more amenable to detect certain attacks but may not be suitable to characterise several other attacks. Most of the existing malware detection approaches use only one (or a selected few) of the aforementioned feature sets which prevent them from detecting a vast majority of attacks. Addressing this limitation, we propose MKLDroid, a unified framework that systematically integrates multiple views of apps for performing comprehensive malware detection and malicious code localisation. The rationale is that, while a malware app can disguise itself in some views, disguising in every view while maintaining malicious intent will be much harder. MKLDroid uses a graph kernel to capture structural and contextual information from apps' dependency graphs and identify malice code patterns in each view. Subsequently, it employs Multiple Kernel Learning (MKL) to find a weighted combination of the views which yields the best detection accuracy. Besides multi-view learning, MKLDroid's unique and salient trait is its ability to locate fine-grained malice code portions in dependency graphs (e.g., methods/classes). Through our large-scale experiments on several datasets (incl. wild apps), we demonstrate that MKLDroid outperforms three state-of-the-art techniques consistently, in terms of accuracy while maintaining comparable efficiency. In our malicious code localisation experiments on a dataset of repackaged malware, MKLDroid was able to identify all the malice classes with 94% average recall

Metaplex networks: influence of the exo-endo structure of complex systems on diffusion

In a complex system the interplay between the internal structure of its entities and their interconnection may play a fundamental role in the global functioning of the system. Here, we define the concept of metaplex, which describes such trade-off between internal structure of entities and their interconnections. We then define a dynamical system on a metaplex and study diffusive processes on them. We provide analytical and computational evidences about the role played by the size of the nodes, the location of the internal coupling areas, and the strength and range of the coupling between the nodes on the global dynamics of metaplexes. Finally, we extend our analysis to two real-world metaplexes: a landscape and a brain metaplex. We corroborate that the internal structure of the nodes in a metaplex may dominate the global dynamics (brain metaplex) or play a regulatory role (landscape metaplex) to the influence of the interconnection between nodes.Comment: 28 pages, 19 figure