## Making Communities Show Respect for Order

Talk given at The Conference on Complex Systems (CCS) Singapore 2019, 1/10/19

Nodes in networks have many natural orders. Every centrality measure allows us to say if one node has a higher centrality value than another. Many real world networks express an important constraint leading to a characteristic order; examples include publication dates of papers in a citation network, dependency of packages in computer software, and prey species in a food web. If edges respect this order, they exist only if they link a high value node to a lower value node, then edges are directed and there can be no cycles - a Directed Acyclic Graph (DAG).

So in a DAG a link between two nodes represents the order of the pair, not necessarily their similarity as assumed in standard network analysis. To understand these network topologies, we need to adapt our tools to make them respect the order implicit in a DAG.

Here we explore a variation of a community detection algorithm which respects such an order and also finds similar nodes. We use the number of common neighbours as our similarity measure. For instance, the greater the overlap in the citations or bibliographies of two publications, the more similar they are, whether or not they cite each other. In most real-world cases, the most comparable nodes would be those that belong to the same hierarchical layer, as defined by the order,

and which share many common neighbours. Nodes in the same hierarchical layer form an \textsc{antichain} so our algorithm finds antichains with a large neighbourhood overlap.

We do this by adapting the Quality function used in Modularity (Newman and Girvan, 2002)

so that it respects the order and uses neighbour overlap to capture similarity. In terms of adjacency matrix A, we optimise the "siblinarity"

S of a node partition as defined in this talk.

We study the algorithm's performance and antichain properties in artificial models and in real networks, such as citation graphs and food webs. We show how well this partitioning algorithm distinguishes and groups together nodes of the same origin (in a citation network, the origin is a topic or a research field) as the figure illustrates. We make the comparison between our partitioning algorithm and standard hierarchical layering tools as well as community detection methods. We show that our algorithm outperforms the standard tools in finding similar, yet hierarchically equivalent sets of nodes.