On convexity in complex networks

Lovro Šubelj University of Ljubljana, Faculty of Computer and Information Science Ljubljana, Slovenia [email protected] Tilen Marc Institute of Mathematics, Physics and Mechanics Ljubljana, Slovenia [email protected] Metric graph theory is a study of geometric properties of graphs based on a notion of the shortest path between the nodes defined as the path through the smallest number of edges [2]. Metric graph properties lie in the heart of the analysis of complex networks. Classical examples include Milgram’s experiment of degrees of separation [8], node index called betweenness centrality [5] and the small-world network model [11]. Independently of these efforts, graph theorists have been interested in understanding convexity in a given graph [6]. Consider a simple connected graph and a subgraph on some subset of nodes S. The subgraph is induced if all edges between the nodes in S in the graph are also included in the subgraph. Next, the subgraph is said to be isometric if at least one shortest path joining each two nodes in S is entirely included within S. Finally, the subgraph is a convex subgraph if all shortest paths between the nodes in S are entirely included within S. Notice that any convex subgraph is also isometric, while any isometric subgraph must necessarily be induced. We study convexity in complex networks through the definition of a convex subgraph [7]. We explore convexity from a local and global perspective by analyzing the frequency of small convex subgraphs and the expansion of randomly grown convex subgraphs. In the case of the latter, we grow random connected subgraphs one node at a time and expand them to convex subgraphs if needed. For instance, every connected subgraph of a tree or a complete graph is convex and thus no expansion occurs. Hence, the expansion of convex subgraphs quantifies the presence of a tree-like or clique-like structure in a network. We demonstrate three distinct forms of convexity in graphs and networks. Global convexity refers to a tree-like or clique-like structure of a network as a whole in which convex subgraphs grow very slowly and thus any connected subgraph is likely to be convex. Globally convex networks are spatial infrastructure networks and network science coauthorship graph. In random graphs [4], however, there is a sudden expansion of convex subgraphs when their size exceeds lnn/ ln 〈k〉 nodes, where n is the number of nodes in a graph and 〈k〉 the average node degree. In fact, the only network studied that is globally less convex than a random graph is the Little Rock food web. On the other hand, random graphs are locally convex meaning that any connected subgraph with up to lnn/ ln 〈k〉 nodes is almost certainly convex. Globally convex networks are also fairly locally convex, or even more convex than random graphs under a loose definition of local convexity, whereas almost any other network studied is locally less convex than a random graph. Still, most of these networks are regionally convex. Regional convexity refers to any type of heterogeneous network structure that is only partly convex. For instance, networks with core-periphery structure can be divided into a non-convex c-core surrounded by a convex periphery. Such are the Oregon Internet map and C. elegans protein network. Note that this type of regional convexity does not necessarily imply local convexity. This 5 10 15 0 0.2 0.4 0.6 0.8 1 Random tree